Functions and Function Evaluation: College Algebra Study Notes

Functions

Definition and Notation

A function is a relation that assigns each element in a set (called the domain) to exactly one element in another set (called the range). Functions are commonly denoted by letters such as f, g, or h, and the value of the function at a particular input x is written as f(x).

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (f(x)-values) produced by the function.

• Function notation: If f is a function and x is in its domain, then f(x) is the output value.

Given Functions

Consider the following functions:

• $f(x) = x$

• $g(x) = |x + 1|$

• $h(x) = \frac{1}{x - 3}$

Each function has a different rule for assigning outputs to inputs:

• f(x): The identity function, which returns the input value.

• g(x): The absolute value function, which returns the non-negative value of x + 1.

• h(x): A rational function, undefined when x = 3 because the denominator becomes zero.

Function Evaluation

To evaluate a function means to substitute a specific value for the variable and simplify.

• For example, to evaluate $f(-2)$, substitute $x = -2$ into $f(x)$:

• $f(-2) = -2$

• To evaluate $g(-2)$, substitute $x = -2$ into $g(x)$:

• $g(-2) = |-2 + 1| = |-1| = 1$

• To evaluate $h(-2)$, substitute $x = -2$ into $h(x)$:

• $h(-2) = \frac{1}{-2 - 3} = \frac{1}{-5}$

Evaluating Composite Functions

A composite function is formed when one function is substituted into another. The notation $(f - g)(x)$ means $f(x) - g(x)$.

• To evaluate $(f - g)(-2)$, calculate $f(-2)$ and $g(-2)$, then subtract:

• $f(-2) = -2$

• $g(-2) = 1$

• So, $(f - g)(-2) = f(-2) - g(-2) = -2 - 1 = -3$

Example Table: Function Evaluation

x	f(x)	g(x)	h(x)	(f - g)(x)
-2	-2	1	-0.2	-3
0	0	1	-0.333...	-1
4	4	5	1	-1

Additional info: Table entries for x = 0 and x = 4 are inferred for further illustration.

Key Properties

• Absolute value: $|a|$ is always non-negative.

• Rational functions: Undefined when the denominator is zero.

• Composite functions: Combine two or more functions using addition, subtraction, multiplication, or division.