1. Elementary Formulas and Basic Graphs

2. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0.

3. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0.

4. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator.
(i.e. 10 degrees)

5. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator. (Ans ≈ 1.13).

6. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator. (Ans ≈ 1.13).
Note that we use the following keys to execute it.
* number–keys * operation–keys +, – , *, /.

7. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator. (Ans ≈ 1.13).
Note that we use the following keys to execute it.
* number–keys * operation–keys +, – , *, /.
* the yx–key
* sin( )
* log( )
The formula–keys

8. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator. (Ans ≈ 1.13).
Note that we use the following keys to execute it.
* number–keys * operation–keys +, – , *, /.
The TI SE Calculator Keyboard
* the yx–key
* sin( )
* log( )
Here is a typical keyboard
layout for a scientific
calculator.
The formula–keys

9. Elementary Formulas and Basic Graphs
Let’s start with evaluating log [ (2x + 1)/(sin1/3(x) + 1) ]
at x = 0. The answer is 0. Now let’s analyze the steps
we would take in the calculation by evaluating at
x = 10o using a scientific calculator. (Ans ≈ 1.13).
Note that we use the following keys to execute it.
* number–keys * operation–keys +, – , *, /.
The TI SE Calculator Keyboard
* the yx–key
* sin( )
* log( )
Here is a typical keyboard
layout for a scientific
calculator. The regions of the
mentioned above are noted.
The formula–keys

10. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method.
Elementary Formulas and Basic Graphs

11. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
Elementary Formulas and Basic Graphs

12. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs

13. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs
The three families of formulas
on every scientific calculator

14. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs
An algebraic formula (in x) is a formula formed with
the variable x, real numbers,
The three families of formulas
on every scientific calculator

15. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs
An algebraic formula (in x) is a formula formed with
the variable x, real numbers, repeated applications of
the algebraic operations +, – , *, /
The three families of formulas
on every scientific calculator

16. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs
An algebraic formula (in x) is a formula formed with
the variable x, real numbers, repeated applications of
the algebraic operations +, – , *, / and raising
fractional powers ( )p/q
The three families of formulas
on every scientific calculator

17. Among the many methods of recording measurements
such as printed tables, digital databases, graphs,
charts etc, using mathematical formulas is definitely
the most efficient method. Most of the mathematics
formulas used in the real world are “composed” with
members from the following three groups of formulas.
* the algebraic family
* the trigonometric family
* the exponential–log family
Elementary Formulas and Basic Graphs
An algebraic formula (in x) is a formula formed with
the variable x, real numbers, repeated applications of
the algebraic operations +, – , *, / and raising
fractional powers ( )p/q (which includes the root–
operations  , or  etc.. entered with the yx – key).3 4
The three families of formulas
on every scientific calculator

18. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,

19. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,

20. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + π)1/3
(4x2 – (x + 4)1/2)1/4

21. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + π)1/3
(4x2 – (x + 4)1/2)1/4
We remind the readers that aNxN + aN-1xN-1...+ a1x + a0
where ai are numbers and N is a non–negative
integer are called polynomials (in x) as 3x2 – 2x + 4.

22. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + π)1/3
(4x2 – (x + 4)1/2)1/4
We remind the readers that aNxN + aN-1xN-1...+ a1x + a0
where ai are numbers and N is a non–negative
integer are called polynomials (in x) as 3x2 – 2x + 4.
The algebraic expressions , where P and Q are
polynomials, are called rational expressions.
P
Q

23. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + π)1/3
(4x2 – (x + 4)1/2)1/4
We remind the readers that aNxN + aN-1xN-1...+ a1x + a0
where ai are numbers and N is a non–negative
integer are called polynomials (in x) as 3x2 – 2x + 4.
The algebraic expressions , where P and Q are
polynomials, are called rational expressions.
P
Q
x2 + 3
3 x3 – 2x – 4
is a rational expression.

24. Elementary Formulas and Basic Graphs
Examples of algebraic expressions are
3x2 – 2x + 4,
x2 + 3
3 x3 – 2x – 4
,
(x1/2 + π)1/3
(4x2 – (x + 4)1/2)1/4
We remind the readers that aNxN + aN-1xN-1...+ a1x + a0
where ai are numbers and N is a non–negative
integer are called polynomials (in x) as 3x2 – 2x + 4.
The algebraic expressions , where P and Q are
polynomials, are called rational expressions.
P
Q
x2 + 3
3 x3 – 2x – 4
x2 + 3
3x3 – 2x – 4
is not because of x3.
is a rational expression.

25. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs

26. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,

27. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses,

28. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,

29. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,
and all their constant multiples the basic formulas.

30. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,
and all their constant multiples the basic formulas.
Recall the “composition” operation i.e. input a formula
f(x) into another formula g(x) and get g(f(x)).

31. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,
and all their constant multiples the basic formulas.
Recall the “composition” operation i.e. input a formula
f(x) into another formula g(x) and get g(f(x)).
E.g. if f(x) = sin(x), g(x) = x2, then g(f(x)) = sin2(x).

32. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,
and all their constant multiples the basic formulas.
Recall the “composition” operation i.e. input a formula
f(x) into another formula g(x) and get g(f(x)).
E.g. if f(x) = sin(x), g(x) = x2, then g(f(x)) = sin2(x).
A formula that may be built from the “basic formulas”
by applying the +, – , *, / and the composition
operations in finitely many steps is called an
elementary formula.

33. The trigonometric formulas such as sin(x), arctan(x)
and the exponential–log formulas such as ex, ln(x) are
called transcendental (non–algebraic) formulas.
Elementary Formulas and Basic Graphs
Let’s call the polynomial formulas, the ( )p/q,
the six trig–formulas and their inverses, logb(x) and bx,
and all their constant multiples the basic formulas.
Recall the “composition” operation i.e. input a formula
f(x) into another formula g(x) and get g(f(x)).
E.g. if f(x) = sin(x), g(x) = x2, then g(f(x)) = sin2(x).
A formula that may be built from the “basic formulas”
by applying the +, – , *, / and the composition
operations in finitely many steps is called an
elementary formula. Basically, scientific calculators
handle calculations of elementary formulas.

34. Elementary Formulas and Basic Graphs
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.

35. Elementary Formulas and Basic Graphs
Example A. Assemble the following formulas from
real numbers and the basic formulas using algebraic
operations: +, – , *, /, or the composition operation.
(All the basic formulas are shown in italic.)
a. 2x2 + cos(x)
b. ln(2x2 + cos(x))
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.

36. Elementary Formulas and Basic Graphs
Example A. Assemble the following formulas from
real numbers and the basic formulas using algebraic
operations: +, – , *, /, or the composition operation.
(All the basic formulas are shown in italic.)
a. 2x2 + cos(x)
b. ln(2x2 + cos(x))
Take 2x2
cos(x)
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.

37. Elementary Formulas and Basic Graphs
Example A. Assemble the following formulas from
real numbers and the basic formulas using algebraic
operations: +, – , *, /, or the composition operation.
(All the basic formulas are shown in italic.)
a. 2x2 + cos(x)
b. ln(2x2 + cos(x))
Take 2x2
cos(x)
+
2x2 + cos(x)
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.

38. Elementary Formulas and Basic Graphs
Example A. Assemble the following formulas from
real numbers and the basic formulas using algebraic
operations: +, – , *, /, or the composition operation.
(All the basic formulas are shown in italic.)
a. 2x2 + cos(x)
b. ln(2x2 + cos(x))
Take 2x2
cos(x)
+
2x2 + cos(x)
2x2
cos(x)
+
2x2 + cos(x)
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.

39. Elementary Formulas and Basic Graphs
Example A. Assemble the following formulas from
real numbers and the basic formulas using algebraic
operations: +, – , *, /, or the composition operation.
(All the basic formulas are shown in italic.)
a. 2x2 + cos(x)
b. ln(2x2 + cos(x))
Take 2x2
cos(x)
+
2x2 + cos(x)
2x2
cos(x)
+
2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
The algebraic construction of a given elementary
formula is reflected in the corresponding successive
operations performed when we evaluate the formula.
u

40. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2

41. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation

42. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
sin(x) eu = esin(x)
composition
operation

43. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
composition
operation

44. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
/ (divide)

45. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
/ (divide)
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
u1/2

46. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
/ (divide)
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
u1/2=
ln(2x2 + cos(x))
esin(x) – x2 ][
1/2

47. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
/ (divide)
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
u1/2=
ln(2x2 + cos(x))
esin(x) – x2 ][
1/2
Almost all the formulas we have in this course are
elementary formulas i.e. they are made in this manner
from basic formulas.

48. Elementary Formulas and Basic Graphs
d.
ln(2x2 + cos(x))
esin(x) – x2[ ]
1/2
2x2
cos(x)
+ 2x2 + cos(x) ln(u) = ln(2x2 + cos(x))
composition
operation
x2
sin(x) eu = esin(x)
– esin(x) – x2
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
/ (divide)
ln(2x2 + cos(x))
esin(x) – x2
composition
operation
u1/2=
ln(2x2 + cos(x))
esin(x) – x2 ][
1/2
Almost all the formulas we have in this course are
elementary formulas i.e. they are made in this manner
from basic formulas. Symbolic techniques in calculus
work precisely because elementary formulas are built
from the basic formulas with the above operations.

49. Elementary Formulas and Basic Graphs
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas.

50. Elementary Formulas and Basic Graphs
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x).

51. Elementary Formulas and Basic Graphs
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1).

52. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x),
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define

53. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.

54. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x)
b. (f g)(x)
c. [3h(x) + 2(g g)(x)]1/2

55. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x) = g( f(x) ) = g( 2x2 + 1)
b. (f g)(x)
c. [3h(x) + 2(g g)(x)]1/2

56. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x) = g( f(x) ) = g( 2x2 + 1) = ln(2x2 + 1)
b. (f g)(x)
c. [3h(x) + 2(g g)(x)]1/2

57. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x) = g( f(x) ) = g( 2x2 + 1) = ln(2x2 + 1)
b. (f g)(x) = f( g(x)) = f( ln(x) )
c. [3h(x) + 2(g g)(x)]1/2

58. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x) = g( f(x) ) = g( 2x2 + 1) = ln(2x2 + 1)
b. (f g)(x) = f( g(x)) = f( ln(x) ) = 2ln2(x) + 1
c. [3h(x) + 2(g g)(x)]1/2

59. Elementary Formulas and Basic Graphs
(g f)(x) = g(f(x)); plug f(x) into g(x), and
It’s useful to use the function notation to keep track of
the “deconstruction” of elementary formulas. That is,
we would name the formulas as f(x), g(x), and h(x)
and perform algebraic operations +, – , * , / using
f(x), g(x) or h(x). Hence if f(x) = 2x2, g(x) = ln(x + 1),
then f(x) + 3g(x) = 2x2 + 3ln(x + 1). Finally, we define
(f g)(x) = f(g(x)); plug g(x) into f(x)
for the composition operation.
Example B. Given that f(x) = 2x2 + 1, g(x) = ln(x),
h(x) = sin(x). Simplify the following.
a. (g f)(x) = g( f(x) ) = g( 2x2 + 1) = ln(2x2 + 1)
b. (f g)(x) = f( g(x)) = f( ln(x) ) = 2ln2(x) + 1
c. [3h(x) + 2(g g)(x)]1/2 = 3sin(x) + 2ln( ln(x) )

60. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).

61. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)

62. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)

63. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.

64. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.
Let f(x) = 9 + x2, g(x) = sin(x),

65. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.
Let f(x) = 9 + x2, g(x) = sin(x),
then (g f)(x) = g(f(x)) = sin(9 + x2).

66. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.
Let f(x) = 9 + x2, g(x) = sin(x),
then (g f)(x) = g(f(x)) = sin(9 + x2).
Let h(x) = 4x3 + 5,

67. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.
Let f(x) = 9 + x2, g(x) = sin(x),
then (g f)(x) = g(f(x)) = sin(9 + x2).
Let h(x) = 4x3 + 5,
then h(g f) = h(sin(9 + x2)) = 4sin3(9 + x2) + 5.

68. Elementary Formulas and Basic Graphs
Example C.
a. Express sin(9 + x2) as the composition of
a polynomial and a trig–function (basic formulas).
Let f(x) = 9 + x2, g(x) = sin(x)
Then (g f)(x) = g(f(x)) = sin(9 + x2)
b. Express 4sin3(9 + x2) + 5 using +, – , * , /, and the
composition operation with basic formulas.
Let f(x) = 9 + x2, g(x) = sin(x),
then (g f)(x) = g(f(x)) = sin(9 + x2).
Let h(x) = 4x3 + 5,
then h(g f) = h(sin(9 + x2)) = 4sin3(9 + x2) + 5.
The calculations in calculus are based on the fact
that the elementary functions are built in steps in this
manner from the basic formulas.

69. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).

70. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
y= f(x)

71. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y

72. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y
Graphs of 1st and 2nd degree functions

73. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y
Graphs of 1st and 2nd degree functions
The graphs of linear functions
y = f(x) = mx + b
are non–vertical lines where
m is the slope, and (0, b) is the y
intercept.

74. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y
Graphs of 1st and 2nd degree functions
The graphs of linear functions
y = f(x) = mx + b
are non–vertical lines where
m is the slope, and (0, b) is the y
intercept. The graph of
y = f(x) = 2x + 6 is drawn here by
plotting the x and y intercepts.

75. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y
Graphs of 1st and 2nd degree functions
The graphs of linear functions
y = f(x) = mx + b
are non–vertical lines where
m is the slope, and (0, b) is the y
intercept. The graph of
y = f(x) = 2x + 6 is drawn here by
plotting the x and y intercepts.
(0, 6)
(–3, 0)
y = 2x + 6

76. Elementary Formulas and Basic Graphs
Set the function output f(x) as y,
the plot of all the points (x, y = f(x))
is the graph of the function f(x).
The coordinate of a generic point on
the graph is (x, f(x)) or (x, y) as shown. x
(x, f(x))
y= f(x)
f(x) or y
Graphs of 1st and 2nd degree functions
The graphs of linear functions
y = f(x) = mx + b
are non–vertical lines where
m is the slope, and (0, b) is the y
intercept. The graph of
y = f(x) = 2x + 6 is drawn here by
plotting the x and y intercepts.
(0, 6)
(–3, 0)
(x, 2x+6)
y = 2x + 6

77. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).

78. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).
Parabolas
, #)
2a
–b(vertex =

79. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).
Parabolas
The parabola opens up if a >0, opens down if a < 0.
, #)
2a
–b(vertex =

80. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).
To plot a 2nd degree function, plot the vertex point at
x = – b/(2a).
Parabolas
The parabola opens up if a >0, opens down if a < 0.
, #)
2a
–b(vertex =

81. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).
To plot a 2nd degree function, plot the vertex point at
x = – b/(2a). Then plot the y–intercept (0, c) and it’s
reflection across the line of symmetry.
Parabolas
The parabola opens up if a >0, opens down if a < 0.
, #)
2a
–b(vertex =

82. Elementary Formulas and Basic Graphs
The graphs of quadratic (2nd degree) functions
y = f(x) = ax2 + bx + c are parabolas with
vertex at x = – b/(2a) and y–intercept is at (0, c).
To plot a 2nd degree function, plot the vertex point at
x = – b/(2a). Then plot the y–intercept (0, c) and it’s
reflection across the line of symmetry. Trace the
parabola with these points.
Parabolas
The parabola opens up if a >0, opens down if a < 0.
, #)
2a
–b(vertex =

83. Elementary Formulas and Basic Graphs
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.

84. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9)
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.

85. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.

86. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.

87. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.
(0, 8)
(–1, 9)
x

88. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.
The reflection of (0, 8) across
the line of symmetry is (–2, 8). (0, 8)
(–1, 9)
(–2 , 8)
x

89. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.
The reflection of (0, 8) across
the line of symmetry is (–2, 8).
Trace the parabola with these
points.
(0, 8)
(–1, 9)
(–2 , 8)
x

90. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.
The reflection of (0, 8) across
the line of symmetry is (–2, 8).
Trace the parabola with these
points.
(0, 8)
(–1, 9)
(–2 , 8)
Finally, we solve f(x) = 0 to find
the x–intercept. x

91. Elementary Formulas and Basic Graphs
It’s a parabola that opens down with vertex at
x = – b/(2a) = –1 or (–1, f(–1) = 9), y–intercept is (0, 8).
Plot these points.
Example D.
Sketch the graph of y = f(x) = – x2 – 2x + 8.
The reflection of (0, 8)across
the line of symmetry is (–2, 8).
Trace the parabola with these
points.
(0, 8)
(–1, 9)
(–2 , 8)
Finally, we solve f(x) = 0 to find
the x–intercept. Hence
–x2 – 2x + 8 = 0 or x = 4, – 2.
Label these points on the graph.
(2, 0)(–4, 0)
x

92. Elementary Formulas and Basic Graphs
Graphs of Simple Algebraic Functions y = f(x)
y = 1
y = x/3 + 1
y = –x + 1
y = –x2 + 2x + 8 y = x2/10 + 8
(–1, 8.01) (1, 8.01)
(0, 8)
y = x3 y = x3 – x
y = x1/2
y = 1/x
Vertical
Asymptote
at x = 0
y = x1/3
Horizontal
Asymptote
at y = 0

93. Elementary Formulas and Basic Graphs
There are two other common operations for functions.

94. Elementary Formulas and Basic Graphs
There are two other common operations for functions.
I. The “Absolute Value Operation”

95. Elementary Formulas and Basic Graphs
There are two other common operations for functions.
I. The “Absolute Value Operation”
The “Absolute Value Operation” applies to individual
function f(x) and changes it into |f(x)|.

96. Elementary Formulas and Basic Graphs
There are two other common operations for functions.
I. The “Absolute Value Operation”
The “Absolute Value Operation” applies to individual
function f(x) and changes it into |f(x)|. The outputs of
|f(x)| are always nonnegative.

97. Elementary Formulas and Basic Graphs
There are two other common operations for functions.
I. The “Absolute Value Operation”
The “Absolute Value Operation” applies to individual
function f(x) and changes it into |f(x)|. The outputs of
|f(x)| are always nonnegative. The graph of |f(x)| is
easily obtained from the graph of f(x) by reflecting all
the negative f(x)’s above the x–axis as shown here.

98. Elementary Formulas and Basic Graphs
y = x2– 4
(2, 0)(–2, 0)
(0, –4)
There are two other common operations for functions.
I. The “Absolute Value Operation”
The “Absolute Value Operation” applies to individual
function f(x) and changes it into |f(x)|. The outputs of
|f(x)| are always nonnegative. The graph of |f(x)| is
easily obtained from the graph of f(x) by reflecting all
the negative f(x)’s above the x–axis as shown here.

99. Elementary Formulas and Basic Graphs
y = x2– 4
(2, 0)(–2, 0)
→
(0, –4)
y = |x2– 4|
(0, 4)
(2, 0)(–2, 0)
There are two other common operations for functions.
I. The “Absolute Value Operation”
The “Absolute Value Operation” applies to individual
function f(x) and changes it into |f(x)|. The outputs of
|f(x)| are always nonnegative. The graph of |f(x)| is
easily obtained from the graph of f(x) by reflecting all
the negative f(x)’s above the x–axis as shown here.
Absolute Value

100. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”

101. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.

102. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,

103. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here.

104. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here. x
y
y = f(x)

105. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here. x
y
y = f(x)

106. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here. x
y
y = f(x)

107. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here. x
y
y = f(x)

108. Elementary Formulas and Basic Graphs
II. The “Cut and Paste Operation”
The “Cut and Paste Operation” pieces many parts
from different functions to make new functions. The
results are known as the piecewise functions.
f(x) =
1 if 0 < x
0 if 0 = x
–1 if x < 0
For example,
f(x) is pieced together with three
parts and it’s graph is drawn here.
We often construct these as
conceptual examples–such an f(x)
as this is not elementary and its
graph is broken in a manner that
we will address later.
x
y
y = f(x)