Lesson 4.8

Evaluating Functions Algebraically

The instructions are in the name. Swap out x, plug in the number, and crunch the math.

Introduction

"Evaluate" just means "Find the value." If you know the rule and you know the input is 5, you can find the output. It's substitution with a new name.

Past Knowledge

Lesson 4.7 (Function Notation) and Order of Operations (PEMDAS).

Today's Goal

Substitute a number into a function rule and simplify.

Future Success

Later, you'll evaluate with variables, like finding . Master the numbers first.

Key Concepts

The "Blank Space" Method

The safest way to evaluate is to rewrite the function with empty parentheses.

Equation

f(x) = 3x - 5

↓

Skeleton

f( ) = 3( ) - 5

↓

Plug In 4

f(4) = 3(4) - 5

Using parentheses prevents sign errors every time.

Worked Examples

Example 1: Linear Function

Basic

Evaluate for .

Substitute

Replace x with (4).

Simplify

Multiply first, then add.

Example 2: Quadratic with Negatives

Intermediate

Evaluate for .

Substitute with Parentheses!

This is critical for the squared term.

Simplify

Recall that $(-5)^2$ is positive 25.

Subtracting a negative becomes adding a positive

Example 3: Evaluating an Expression

Advanced

Evaluate if .

Step 1: Find f(3)

Step 2: Find f(5)

Step 3: Add results

Common Pitfalls

Squaring Negatives

Without parentheses, you calculator might say . But . Always use parentheses when plugging in negative numbers.

Stopping Too Early

Some students write and stop. You must do the arithmetic to find the final single value.

Real-Life Applications

Currency Conversion: If converts Dollars to Euros, you evaluate to know how many Euros you get for $100. This substitution is happening every time you swipe a card abroad.