Functions are the fundamental concept in calculus. We discuss four ways to represent a function and different classes of mathematical expressions for functions.
1. Section 1.1–1.2
Functions
V63.0121.006/016, Calculus I
New York University
May 17, 2010
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. . . . . .
2. Announcements
Get your WebAssign
accounts and do the Intro
assignment. Class Key:
nyu 0127 7953
Office Hours: TBD
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 2 / 54
3. Objectives: Functions and their Representations
Understand the definition
of function.
Work with functions
represented in different
ways
Work with functions
defined piecewise over
several intervals.
Understand and apply the
definition of increasing and
decreasing function.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 3 / 54
4. Objectives: A Catalog of Essential Functions
Identify different classes of
algebraic functions,
including polynomial
(linear, quadratic, cubic,
etc.), polynomial
(especially linear,
quadratic, and cubic),
rational, power,
trigonometric, and
exponential functions.
Understand the effect of
algebraic transformations
on the graph of a function.
Understand and compute
the composition of two
functions. . . . . . .
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5. What is a function?
Definition
A function f is a relation which assigns to to every element x in a set D
a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 5 / 54
6. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 6 / 54
7. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 7 / 54
9. The Modeling Process
. .
Real-world
.
. m
. odel Mathematical
.
Problems Model
s
. olve
.est
t
. i
.nterpret .
Real-world
. Mathematical
.
Predictions Conclusions
S
. hadows F
. orms
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 9 / 54
10. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 10 / 54
11. Functions expressed by formulas
Any expression in a single variable x defines a function. In this case,
the domain is understood to be the largest set of x which after
substitution, give a real number.
. . . . . .
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12. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
. . . . . .
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13. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
14. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2. As for the range, we can solve
x+2 2y + 1
y= =⇒ x =
x−1 y−1
So as long as y ̸= 1, there is an x associated to y.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
15. Formula function example
Example
x+1
Let f(x) = . Find the domain and range of f.
x−2
Solution
The denominator is zero when x = 2, so the domain is all real numbers
except 2. As for the range, we can solve
x+2 2y + 1
y= =⇒ x =
x−1 y−1
So as long as y ̸= 1, there is an x associated to y. Therefore
domain(f) = { x | x ̸= 2 }
range(f) = { y | y ̸= 1 }
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 12 / 54
16. No-no's for expressions
Cannot have zero in the denominator of an expression
Cannot have a negative number under an even root (e.g., square
root)
Cannot have the logarithm of a negative number
. . . . . .
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17. Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.
. . . . . .
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18. Piecewise-defined functions
Example
Let {
x2 0 ≤ x ≤ 1;
f(x) =
3−x 1 < x ≤ 2.
Find the domain and range of f and graph the function.
Solution
The domain is [0, 2]. The range is [0, 2). The graph is piecewise.
. .
2 .
. .
1 . .
. . .
0
. 1
. 2
.
. . . . . .
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19. Functions described numerically
We can just describe a function by a table of values, or a diagram.
. . . . . .
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20. Example
Is this a function? If so, what is the range?
x f(x)
1 4
2 5
3 6
. . . . . .
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21. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 16 / 54
22. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 5
3 6
. .
3 ..
6
Yes, the range is {4, 5, 6}.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 16 / 54
23. Example
Is this a function? If so, what is the range?
x f(x)
1 4
2 4
3 6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
24. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
25. Example
Is this a function? If so, what is the range?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
2 4
3 6
. .
3 ..
6
Yes, the range is {4, 6}.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 17 / 54
26. Example
How about this one?
x f(x)
1 4
1 5
3 6
. . . . . .
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27. Example
How about this one?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 18 / 54
28. Example
How about this one?
. .
1 ..
4
x f(x)
1 4 . ..
2 .. .
5
1 5
3 6
. .
3 ..
6
No, that one’s not “deterministic.”
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 18 / 54
30. Why numerical functions matter
In science, functions are often defined by data. Or, we observe data
and assume that it’s close to some nice continuous function.
. . . . . .
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31. Numerical Function Example
Here is the temperature in Boise, Idaho measured in 15-minute
intervals over the period August 22–29, 2008.
.
1
. 00 .
9
.0.
8
.0.
7
.0.
6
.0.
5
.0.
4
.0.
3
.0.
2
.0.
1
.0. . . . . . . .
8
. /22 . /23 . /24 . /25 . /26 . /27 . /28 . /29
8 8 8 8 8 8 8
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 21 / 54
32. Functions described graphically
Sometimes all we have is the “picture” of a function, by which we
mean, its graph.
.
.
. . . . . .
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33. Functions described graphically
Sometimes all we have is the “picture” of a function, by which we
mean, its graph.
.
.
The one on the right is a relation but not a function.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 22 / 54
34. Functions described verbally
Oftentimes our functions come out of nature and have verbal
descriptions:
The temperature T(t) in this room at time t.
The elevation h(θ) of the point on the equator at longitude θ.
The utility u(x) I derive by consuming x burritos.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 23 / 54
35. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 24 / 54
36. Monotonicity
Example
Let P(x) be the probability that my income was at least $x last year.
What might a graph of P(x) look like?
. . . . . .
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37. Monotonicity
Example
Let P(x) be the probability that my income was at least $x last year.
What might a graph of P(x) look like?
. .
1
. .5 .
0
. . .
$
.0 $
. 52,115 $
. 100K
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 25 / 54
38. Monotonicity
Definition
A function f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for
any two points x1 and x2 in the domain of f.
A function f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for any
two points x1 and x2 in the domain of f.
. . . . . .
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39. Examples
Example
Going back to the burrito function, would you call it increasing?
. . . . . .
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40. Examples
Example
Going back to the burrito function, would you call it increasing?
Example
Obviously, the temperature in Boise is neither increasing nor
decreasing.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 27 / 54
41. Symmetry
Example
Let I(x) be the intensity of light x distance from a point.
Example
Let F(x) be the gravitational force at a point x distance from a black
hole.
. . . . . .
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42. Possible Intensity Graph
y
. = I(x)
.
x
.
. . . . . .
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43. Possible Gravity Graph
y
. = F(x)
.
x
.
. . . . . .
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44. Definitions
Definition
A function f is called even if f(−x) = f(x) for all x in the domain of f.
A function f is called odd if f(−x) = −f(x) for all x in the domain of
f.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 31 / 54
46. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 33 / 54
47. Classes of Functions
linear functions, defined by slope an intercept, point and point, or
point and slope.
quadratic functions, cubic functions, power functions, polynomials
rational functions
trigonometric functions
exponential/logarithmic functions
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 34 / 54
48. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
49. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
50. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.
Answer
If x is in miles and f(x) in dollars,
f(x) = 2.5 + 2x
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 35 / 54
51. Example
Biologists have noticed that the chirping rate of crickets of a certain
species is related to temperature, and the relationship appears to be
very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F
and 173 chirps per minute at 80 ◦ F.
(a) Write a linear equation that models the temperature T as a function
of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the
temperature.
. . . . . .
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53. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
. . . . . .
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54. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
55. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
56. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 37 / 54
57. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0, downward if
a < 0.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 38 / 54
58. Other Polynomial functions
Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0, downward if
a < 0.
Cubic functions take the form
f(x) = ax3 + bx2 + cx + d
. . . . . .
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59. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
. . . . . .
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60. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c.
. . . . . .
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61. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 22 + b · 2 + c
0 = a · 32 + b · 3 + c
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
62. Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?
Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:
3 = a · 02 + b · 0 + c
−1 = a · 22 + b · 2 + c
0 = a · 32 + b · 3 + c
Right away we see c = 3. The other two equations become
−4 = 4a + 2b
−3 = 9a + 3b
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 39 / 54
63. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
. . . . . .
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64. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
65. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute
a = 1 into the first equation and we have
−12 = 12 + 6b =⇒ b = −4
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
66. Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a + 6b
−6 = 18a + 6b
Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute
a = 1 into the first equation and we have
−12 = 12 + 6b =⇒ b = −4
So our equation is
y = x2 − 4x + 3
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 40 / 54
67. Other power functions
Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 41 / 54
68. Rational functions
Definition
A rational function is a quotient of polynomials.
Example
x3 (x + 3)
The function f(x) = is rational.
(x + 2)(x − 1)
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 42 / 54
69. Trigonometric Functions
Sine and cosine
Tangent and cotangent
Secant and cosecant
. . . . . .
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70. Exponential and Logarithmic functions
exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example f(x) = log2 (x))
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 44 / 54
71. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 45 / 54
72. Transformations of Functions
Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 46 / 54
73. Transformations of Functions
Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the function, a transformation
is applied on the x-axis. After the function, to the y-axis.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 46 / 54
74. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
75. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
76. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
77. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
78. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units to the left
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 47 / 54
79. Now try these
y = sin (2x)
y = 2 sin (x)
y = e−x
y = −ex
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 48 / 54
80. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f by c
y = c · f(x), scale the graph of f by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
81. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
82. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
83. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
84. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a flip
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 49 / 54
85. Outline
.
Modeling
Examples of functions
Functions expressed by formulas
Functions described numerically
Functions described graphically
Functions described verbally
Properties of functions
Monotonicity
Symmetry
Classes of Functions
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 50 / 54
86. Composition is a compounding of functions in
succession
g
. ◦f
.
x
. f
. . g
. . g ◦ f)(x)
(
f
.(x)
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 51 / 54
87. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 52 / 54
88. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 52 / 54
89. Decomposing
Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?
Solution
√
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f. To
insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 53 / 54
90. Summary
The fundamental unit of investigation in calculus is the function.
Functions can have many representations
There are many classes of algebraic functions
Algebraic rules can be used to sketch graphs
. . . . . .
V63.0121.006/016, Calculus I (NYU) Section 1.1–1.2 Functions May 17, 2010 54 / 54