Evaluating Functions and Function Notation

Introduction

Understanding how to evaluate functions and use function notation is a foundational skill in precalculus. This topic involves substituting specific values into a function's formula and simplifying the result. Mastery of these concepts is essential for further study in mathematics, including calculus and applied sciences.

Function Notation

• Function notation is a way to represent functions using symbols such as f(x), g(x), etc.

• The variable inside the parentheses (usually x) is called the input or independent variable.

• The value of the function for a specific input is found by substituting the input value into the function's formula.

Evaluating Functions

To evaluate a function at a given value, replace the variable in the function's formula with the specified number and simplify.

• Example: If , then .

• Example: If , then .

Key Steps for Evaluating Functions

1. Identify the function and the value to substitute.

2. Replace every occurrence of the variable in the function's formula with the given value.

3. Simplify the resulting expression using the order of operations.

Worked Examples from the Questions

• 1) Find when . Substitute into : Answer:

• 2) If , find . Substitute into : Answer:

• 3) If , find . Substitute into : Answer:

• 4) If , find . Substitute into : Answer:

• 5) Find when . Substitute into : Answer:

• 6) Find when . Substitute into : Answer: Note: The provided answer in the image is , which does not match the expanded result. The correct answer is as shown above.

Summary Table: Function Evaluation Results

Applications and Importance

• Function evaluation is used in modeling real-world situations, such as calculating profit, population growth, or physical quantities.

• Understanding how to manipulate and evaluate functions is essential for solving equations, graphing, and calculus.