Functions and Function Evaluation: College Algebra Review

Functions

Definition of a Function

A function is a relation that assigns exactly one output value for each input value from a given set. Functions are commonly denoted as f(x), g(x), etc., where x is the input variable.

• 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.

Given Functions

• $f(x) = x^2 + 7$

• $g(x) = \sqrt{x} - 2$

• $h(x) = 3x - 7$

Function Evaluation

Evaluating Functions

To evaluate a function at a specific value, substitute the given value for the variable x in the function's formula and simplify.

• Example: To evaluate $f(2)$ for $f(x) = x^2 + 7$:

  • Substitute $x = 2$ into the function: $f(2) = (2)^2 + 7$

  • Simplify: $f(2) = 4 + 7 = 11$

Composite Functions

A composite function is formed when one function is evaluated inside another. The notation $(g \circ f)(x)$ means $g(f(x))$.

• Example: $(g \circ f)(x) = g(f(x))$

• First, evaluate $f(x)$, then use that result as the input for $g(x)$.

Sample Problem

Evaluate the number $h(g(2))$.

1. First, evaluate $g(2)$:

   • $g(x) = \sqrt{x} - 2$

   • $g(2) = \sqrt{2} - 2$

2. Next, use this result as the input for $h(x)$:

   • $h(x) = 3x - 7$

   • $h(g(2)) = 3(\sqrt{2} - 2) - 7$

   • Expand: $= 3\sqrt{2} - 6 - 7$

   • Combine like terms: $= 3\sqrt{2} - 13$

Final Answer: $h(g(2)) = 3\sqrt{2} - 13$

Summary Table: Function Evaluation Steps

Step	Description	Example
1	Evaluate the inner function	$g(2) = \sqrt{2} - 2$
2	Substitute result into the outer function	$h(g(2)) = 3(\sqrt{2} - 2) - 7$
3	Simplify the expression	$3\sqrt{2} - 13$