Calculus 1 Lecture Notes (Functions and Their Graphs)

Calculus 1 Lecture Notes
Functions and Their Graphs
Dr. Mohammed A. Matar
September 27, 2022

CHAPTER 1 Functions
1 Functions and Their Graphs
What exactly is a function?
y=f(x) (“y equals f of x”)
The symbol f represents the function, the letter x is the independent variable representing the input value
to f, and y is the dependent variable or output value of f at x.
Definition:
A function f from a set D to a set Y is a rule that assigns a unique value f(x) in Y to each x in D.
i.e. An equation will be a function if for any x in the domain of the equation (the domain is all the x’s that
can be plugged into the equation) the equation will yield exactly one value of y.
Figure 1: Functions or NOT
Notes:
• The set D of all possible input values is called the domain of the function.
• The set of all output values of f(x) as x varies throughout D is called the range of the function.
• The range might not include every element in the set Y(co-domain).
• Often a function is given by a formula that describes how to calculate the output value from the input
variable.
Natural domain of f: When we define a function y = f(x) with a formula and the domain is not stated
explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the
formula gives real y-values.
To restrict the domain of the function to, say, positive values of x, we would write “y = x2
, x > 0”.
Example 1.1
Determine if each of the following are functions.
1. y = x2
+ 1
2. y2
= x + 1
Dr. Mohammed A. Matar UCAS Page 2 of 12

CHAPTER 1 Functions
Solution
1. The first one is a function. Given an x there is only one way to square it and then add 1 to the result and
so no matter what value of x you put into the equation there is only one possible value of y.
2. The only difference between this equation and the first is that we moved the exponent off the x and onto
the y. This small change is all that is required, in this case, to change the equation from a function to
something that isn’t a function.
To see that this isn’t a function is fairly simple. Choose a value of x, say x = 3 and plug this into the
equation.
y2
= 3 + 1 = 4
Now, there are two possible values of y that we could use here. We could use y = 2 or y = −2 . Since
there are two possible values of y that we get from a single x this equation is not a function.
Note that this only needs to be the case for a single value of x to make an equation not be a function. For
instance we could have used x = −1 and in this case we would get a single y (y = 0). However, because of what
happens at x = 3 this equation will not be a function.
Example 1.2
Let D = {1, 2, 3, 4} and Y = {a, b, c, d}, and define f(1) = a , f(2) = b ,f(3) = b ,f(4) = c. Is f a function or not?
Solution
As can be seen from the figure below, every element in D has an unique output in Y , so we can observe that f
is a function with domain D and co-domain Y and range {a, b, c}
Now, how do we actually evaluate the function? That’s really simple. Everywhere we see an x on the
right side we will substitute whatever is in the parenthesis on the left side. Let’s take a look at some function
evaluation.
Example 1.3
Given f(x) = −x2
+ 6x − 11 find each of the following. Find:
(a) f(2)
(b) f(−10)
(c) f(t)
(d) f(t − 3)
(e) f(x − 3)
(f) f(4x − 1)

CHAPTER 1 Functions
Solution
(a) f(2) = −(2)2
+ 6(2) − 11 = −3
(b) f(−10) = −(−10)2
+ 6(−10) − 11 = −100 − 60 − 11 = −171
Be careful when squaring negative numbers!
(c) f(t) = −t2
+ 6t − 11
Remember that we substitute for the x′
s WHATEVER is in the parenthesis on the left. Often this will
be something other than a number. So, in this case we put t′
s in for all the x′
s on the left.
(d) f(t − 3) = −(t − 3)2
+ 6(t − 3) − 11 = −t2
+ 12t − 38
(e) f(x − 3) = −(x − 3)2
+ 6(x − 3) − 11 = −x2
+ 12x − 38
(f) f(4x − 1) = −(4x − 1)2
+ 6(4x − 1) − 11 = −16x2
+ 32x − 18
In the next example we will verify the natural domains and associated ranges of some simple functions.
Example 1.4
Find the domain and the range of the following functions:
(a) y = x2
(b) y = 1
x
(c) y =
√
x
(d) y =
√
4 − x
(e) y =
√
1 − x2
Solution
(a) The formula y = x2
gives a real y-value for any real number x, so the domain is (−∞, ∞).
−∞ < x < ∞
−∞ < x ≤ 0 and 0 ≤ x < ∞
0 ≤ |x| < ∞ ⇒ 0 ≤ x2
< ∞ ⇒ 0 ≤ y < ∞
The range of the function is [0, ∞).
(b) The formula y = 1
x gives a real y-value for every x except x = 0. The domain of the function is
(−∞, 0) ∪ (0, ∞), and the range of it is (−∞, 0) ∪ (0, ∞).
(c) The formula y =
√
x gives a real y-value only if x ≥ 0. The range of y =
√
x is [0, ∞) because every
non-negative number is some number’s square root (namely, it is the square root of its own square).
(d) In y =
√
4 − x, the quantity 4 − x cannot be negative ⇒ (x ≤ 4). The formula gives non-negative real
y-values for all (x ≤ 4), so that the range is [0, ∞).
(e) Here, 1 − x2
≥ 0 ⇒ x2
≤ 1 ⇒ |x| ≤ 1 ⇒ −1 ≤ x ≤ 1 so that the domain of the function is [−1, 1]. The
rang of the function can be find as follow:
|x| ≤ 1 ⇒ 0 ≤ x2
≤ 1 ⇒ 0 ≥ −x2
≥ 1 ⇒ ⇒ 0 ≤ 1 − x2
≤ 1 ⇒ 0 ≤
√
1 − x2 ≤ 1
so that the range of the function is [0, 1].
###

CHAPTER 1 Functions
Notes:
In order to find the natural domains of a function, we need to consider the following points
• We cannot divide any number by zero.
• When n is even number, we cannot solve the n-root of a negative values.
1.1 Graphs of Functions
If f is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are
the input-output pairs for f. In set notation, the graph is
{(x, f(x)) | x ∈ D}
Example 1.5
Graph the following functions
(a) f(x) = x + 2
(b) y = x2
over the interval [-2, 2].
Solution
(a) The graph of the function ƒ(x) = x + 2 is the set of points with coordinates (x, y) for which y = x + 2.
Its graph is the straight line sketched in the next figure.
In order to find the x-axis crossing points, we can solve the function at y = 0.
y = x + 2 ⇒ 0 = x + 2 ⇒ x = −2 ⇒ (-2 , 0)
By using the same concept, we can find the y-axis crossing points by equaling x to zero.
y = x + 2 ⇒ y = 0 + 2 ⇒ y = 2 ⇒ (0 , 2)

CHAPTER 1 Functions
(b) In this example, we can first make a table of xy-pairs that satisfy the equation y = x2
and then plot the
points (x, y) whose coordinates appear in the table, and draw a smooth curve.
x y
-2 4
-1 1
0 0
1 1
2 4
How we can make the curve more smooth? and how to connect between the points?
1.2 The Vertical Line Test for a Function
A function f can have only one value f(x) for each x in its domain, so no vertical line can intersect the
graph of a function more than once.
Example 1.6
Use the vertical line test to determine if the following rules represent functions or NOT.
(a) x2
+ y2
= a2
(b) y = x2
(c) x = y2
Solution
In this example we will plot the function using the previous method as in next figure.
(a) x2
+ y2
= a2
(b) y = x2
(c) x = y2

CHAPTER 1 Functions
1.3 Piecewise-Defined Functions
Sometimes a function is described in pieces by using different formulas on different parts of its domain. One
example is the absolute value function.
|x| =

x : x ≥ 0 First formula
−x : x ≤ 0 Second formula
The right-hand side of the equation means that the function equals x if x ≥ 0, and equals -x if x ≤ 0.
The domain of the absolute value function Dom(f) = ℜ and the range Ran(f) = [0, ∞)
Remarks
• | − x| = |x|
•
√
x2 = |x| ∀x ∈ ℜ
• |xy| = |x||y|
• |x + y| ≤ |x| + |y| (Triangle inquality)
• |x/y| = |x|
|y|
• Geometrically |x − y| is the distance between x and y.
Example 1.7
The below function is defined on the entire real line but has values given by different formulas, depending on
the position of x.
f(x) =



−x : x 0 First formula
x2
: 0 ≤ x ≤ 1 Second formula
1 : x 1 Third formula

CHAPTER 1 Functions
The function, however, is just one function whose domain is the entire set of real number. What is the
Rang ??
Example 1.8
The function whose value at any number x is the greatest integer less than or equal to x is called the greatest
integer function or the integer floor function. It is denoted ⌊x⌋.
⌊2.4⌋ = 2 , ⌊2⌋ = 2 , ⌊1.9⌋ = 1 , ⌊0.2⌋ = 0 , ⌊−0.3⌋ = −1 , ⌊−1.2⌋ = −2
The function whose value at any number x is the smallest integer greater than or equal to x is called the
least integer function or the integer ceiling function. It is denoted ⌈x⌉.
1.4 Increasing and Decreasing Functions
If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing.
If the graph descends or falls as you move from left to right, the function is decreasing.
Definition
Let f be a function defined on an interval I and let x1 and x2 be two distinct points in I.
1. If f(x2) f(x1) whenever x1 x2, then f is said to be increasing on I.
2. If f(x2) f(x1) whenever x1 x2, then f is said to be decreasing on I.

CHAPTER 1 Functions
Example 1.9
The function is decreasing on (∞, 0) and increasing on (0, 1). The function is neither increasing nor decreasing
on the interval (1, ∞) because the function is constant on that interval.
1.5 Even Functions and Odd Functions: Symmetry
The graphs of even and odd functions have special symmetry properties.
Definition A function y = f(x) is an
even function of x if f(−x) = (x),
odd function of x if f(−x) = −f(x),
for every x in the function’s domain.
Example 1.10
Determine whether the following functions are even, odd, or neither.
(a) y = x2
(b) y = 2x3
(c) y = x3
|x|+1
(d) y = |x − 1|
Solution
(a) f(−x) = (−x)2
= x2
= f(x) ⇒ EVEN
(b) f(−x) = 2(−x)3
= −2x3
= −f(x) ⇒ ODD
(c) f(−x) = (−x)3
|−x|+1 = −x3
|x|+1 = −f(x) ⇒ ODD
(d) f(−x) = | − x − 1| = | − (x + 1)| = (x + 1) ̸= f(x) ̸= f(−x) ⇒ Neither even neither odd
Note
• The graph of an even function is symmetric about the y-axis.
• The graph of an odd function is symmetric about the origin.

CHAPTER 1 Functions
(a) y = x2
(b) y = x3
1.6 Common Functions
A variety of important types of functions are frequently encountered in calculus.
1.6.1 Linear Functions
A function of the form f(x) = mx + b, where m and b are fixed constants, is called a linear function.
• The function f(x) = x where m = 1 and b = 0 is called the identity function.
• Constant functions result when the slope is m = 0.
1.6.2 Power Functions
A function f(x) = xa
, where a is a constant, is called a power function. There are several important cases to
consider.
Note
1. If a ∈ N , as x2
, x3
, x4
, x5
then the Dom(f) = R and the Range of the function is
Ran(f) =

[0, ∞) : a is even
R : a is odd
2. If −a ∈ N , as x−2
, x−3
, x−4
, x−5
then the Dom(f) = R − {0}

CHAPTER 1 Functions
Some Special Cases
(a) f(x) = xa
with a = n, a positive integer.
(b) f(x) = xa
with a = −1, or a = −2.
(c) a = 1
2 , 1
3 , 3
2 , and 2
3
Figure 4: Graphs of f(x) = xn
, n = 1, 2, 3, 4, 5, defined for −∞ x ∞
Figure 5: Graphs of the power functions f(x) = xa
. (a) a = −1, (b) a = −2.
Figure 6: Graphs of the power functions f(x) = xa
a = 1
2 , 1
3 , 3
2 , and 2
3

CHAPTER 1 Functions
1.6.3 Polynomials
Let n be non-negative integer and let a0, a1, a2, ...an be constant real numbers, then a function of the form:
p(x) = anxn
+ an−1xn−1
+ ... + a1x + a0
is called a polynomial function and the constants ai where i = 1, 2, 3, ..., n are the coefficients of the polynomial.
If an ̸= 0, then the degree of the polynomial is equal to n.
Remark: The domain of all polys. is equal to (−∞, ∞).
Example 1.11
(a) y = c where c is constant is a constant poly.
(b) y = mx + b is a linear poly.
(c) y = ax2
+ bx + c is a quadratic poly.
(d) y = ax3
+ bx2
+ cx + d is a cubic poly.
1.6.4 Rational Functions
A rational function is a function of the form f(x) = p(x)
q(x) , where p(x) and q(x) are polys.
Remark: The domain of rational functions is R − {x : q(x) = 0}
Example 1.12
1. y = 1
x is a rational function with a domain R − {0}
2. The function f(x) = x2
−5
x+1 , g(x) = 4x3
−x+1
3x5−2x , and h(x) = −x2
are all rational functions.
1.6.5 Algebraic Functions
Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication,
division, and taking roots) lies within the class of algebraic functions. For example: y = x1/3
(x − 4)
1.6.6 Trigonometric Functions
The six basic trigonometric functions will be covered in Section 1.3.
1.6.7 Exponential and Logarithmic Functions
A function of the form f(x) = ax
, where a 0 and a ̸= 1, is called an exponential function (with base a). All
exponential functions have domain (−∞, ∞) and range (0, ∞), so an exponential function never assumes the
value 0. The Logarithmic functions are the functions f(x) = logax, where the base a ̸= 1 is a positive constant.
They are the inverse functions of the exponential functions
1.6.8 Transcendental Functions
A function which is not algebraic is called transcendental function. They include the trigonometric, inverse
trigonometric, exponential, and logarithmic functions, and many other functions as well. For example: y = cosx,
y = sinx, and y = x2
+ 2x − sinx
THE END

Calculus 1 Lecture Notes (Functions and Their Graphs)

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