Math 1330 Section 1.1
Section 1.1
An Introduction to Functions
Most of this course will deal with functions. Suppose we start with two sets, A and B. A function is a rule which assigns
one and only one element of set B to each element in set A. Set A is called the domain of the function and set B is called
the range.
Example 1: State whether or not the mappings represents a function. If it does, identify its domain and range.
A
B
X
1
Y
2
Z
3
A
B
A
10
B
15
C
20
Functions are usually written using function notation. If an equation is solved for y, such as y  mx  b , we would write
this using function notation as f ( x)  mx  b, read “f of x,” denoting the value of the function at x. We can also use
other lower case letters to denote a function, such as g, h, j, k, etc.
Often, you will be asked to state the domain of a stated function. Domain is a subset of the set of real numbers. We
state domain using interval notation, so we’ll start with a review of that:
Reminder: Interval Notation
(-3, 5)
all x such that
3  x  5
[-3, 5]
all x such that
3  x  5
[-3, 5)
all x such that
3  x  5
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Math 1330 Section 1.1
[3, )
all x such that
x  3
(,5)
all x such that
x 5
(, )
all real numbers
Example 2: Express using interval notation: 3 < 𝑥 ≤ 5
Example 3: Express using interval notation: x  3.
Example 4: Express using interval notation: x  7.
Next, we’ll state the domain of several types of functions.
The domain of any polynomial function is  ,, or all real numbers.
The domain of any rational function, where both the numerator and the denominator are polynomials, is all real
numbers except the values of x for which the denominator equals 0.
The domain of any radical function with even index (2, 4, etc) is the set of real numbers for which the radicand is
greater than or equal to 0. Example would be √𝑥 − 2 and 𝑥 − 2 ≥ 0, solve this to find the domain. The domain of any
radical function with odd index (3,5,etc) is  ,.
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Math 1330 Section 1.1
Example 5: State the domain of the function. Write your answer using interval notation.
g ( x) 
x2  5x  4
x 2  16
Example 6: State the domain of the function. Write your answer using interval notation.
𝑔(𝑥) = √4 − 𝑥
Example 7: State the domain of the function. Write your answer using interval notation.
f ( x) 
x 1
x5
Example 8: State the domain of the function. Write your answer using interval notation.
h( x )  3 x 2  9
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Math 1330 Section 1.1
Example 9: State the domain of the function. Write your answer using interval notation.
f (x) 
x 5
x2
If f ( x)  y , you can solve y for x and use the methods for finding domain to state the range of the function.
Example 10: Find the domain and range of the function: g ( x) 
x2
x5
You also need to be able to evaluate a function at a given value of x or at an expression.
Example 11: If f ( x)  x 2  3x  1 , find 𝑓(−1) and f (t  3).
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Math 1330 Section 1.1
1
Example 12: Find 𝑓 (− 2) given 𝑓(𝑥)
=
2𝑥 2
3𝑥+2
− 5𝑥
x  1
2 x  3,
 2
Example 13: If f ( x)   x  2 x  3,  1  x  3, find , f (3) , f (0) , f (5) and f (3).
 6 x,
x3

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Math 1330 Section 1.1
Next, you will need to be able to form a difference quotient. To find a difference quotient, you will compute
f ( x  h)  f ( x )
, assuming that h  0. You can do this in three steps:
h
Compute f ( x  h).
Then compute f ( x  h)  f ( x).
Then compute
f ( x  h)  f ( x )
.
h
Example 14: Find the difference quotient: f ( x)  x 2  2 x  9
Example 15: Find the difference quotient:
𝑓(𝑥) =
1
5𝑥
6