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 $f(x)$: $f(2) = (2)^2 + 7 = 4 + 7 = 11$

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: If $f(x) = x^2 + 7$ and $g(x) = \sqrt{x} - 2$, then $g(f(2))$ means first evaluate $f(2)$, then substitute that result into $g(x)$.

Step 1: $f(2) = 11$ Step 2: $g(11) = \sqrt{11} - 2$

Exact Form vs. Decimal Form

• Exact form: Leave answers in terms of radicals or fractions (e.g., $\sqrt{11} - 2$).

• Decimal form: Approximate the value using a calculator (e.g., $\sqrt{11} \approx 3.317$, so $3.317 - 2 = 1.317$).

Sample Problem

Evaluate the number $g(f(2))$ in exact form:

• Given $f(x) = x^2 + 7$ and $g(x) = \sqrt{x} - 2$

• First, find $f(2)$:

$f(2) = (2)^2 + 7 = 4 + 7 = 11$

• Next, substitute into $g(x)$:

$g(f(2)) = g(11) = \sqrt{11} - 2$

• Final Answer (exact form): $\sqrt{11} - 2$

Summary Table: Function Evaluation Steps