Section 2.4A: Functions.
Functions are usually denoted as f (x). For this example, f is the name of the function, and x is the independent variable. When plotting a function, f (x) is denoted as the value obtained for x and is usually designated along the y-axis.
Evaluating Functions: Given a function f (x), we can evaluate it for specific (or arbitrary) values of x.
Example: Let f (x) = 8x2 − 1. Compute the following.
3.) 1/f (b)
1.) 5f (a)
2.) f 1b
√
4.) f ( c)
5.)
p
f (c)
Difference Quotient: The Difference Quotient is an algebra tool that you will encounter very early on in Calculus.
Given a function f (x), the difference quotient is:
f (x + h) − f (x)
h
Your task will be to simplify the difference quotient so that it makes sense when h = 0.
Examples: Compute the difference quotient for the following functions:
f (x) = 8x − 1
g(x) = 5x2 − 4
k(x) = 2/(3x)
Domains of Functions: Another topic you will need to know about functions is the concept of Domain. In general,
the domain of a function f (x) are all the permissible real values x that can be input into your function. Similarly,
the range of a function f (x) are all the permissible real values y that can be output out of your function.
Graphical Approach: Given the graph of a function f (x), the domain of f (x) is all the set of x values and the
range of f (x) is all the set of y values.
Examples: Use the Graphical Approach to determine the domain and range of the following functions:
Algebraic Approach: The domain of a function f(x) can be computed algebraically. This is done by considering two
main concepts of algebra:
1. Division by Zero is never allowed in mathematics. By isolating and analyzing any potential x values that violate
this rule, you can determine which values to throw out of your real number line. For functions with denominators,
set the denominator = 0.
√
2. Even Radicals, such as square roots, require a non-negative radicand (the value inside). Note that 0 is zero,
which is a real number. Also, odd roots do not have domain issues. For functions with even radicals, set the
radicant greater than or equal to zero.
Examples: Use the Algebraic Approach to determine the domain of the following functions:
√
√
f (x) = x3 + 4x2 − 6x + 9
g(x) = 5 x2 + 9x − 1
h(x) = 9 − 5x + x2 − 3
j(x) =
5x − 3
2x + 9
k(x) =
p(x) =
9x2 + 8
x2 + 9
5x − 7
18 − 3x2
√
l(x) =
9x − 4
q(x) = √
5x − 11
10 − 3x
x+2