AP Precalculus Practice Test: Unit 1 Question #17 Validating an Even Function

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

In AP Precalculus, Unit 1 includes exploring the properties of functions, including determining whether a function is even. Question #17 on validating an even function likely asks students to determine if a given function \( f(x) \) is even by checking its symmetry properties.

Here's the approach for validating an even function:

1. **Understand the Definition**: A function \( f(x) \) is considered even if \( f(-x) = f(x) \) for all \( x \) in the domain of \( f \). This means the function is symmetric with respect to the y-axis.

2. **Substitute \( -x \) for \( x \)**: To check if a function is even, replace \( x \) with \( -x \) in the function and simplify. If the result is identical to the original function \( f(x) \), the function is even.

3. **Compare Results**: After simplifying \( f(-x) \), if \( f(-x) = f(x) \), then the function is even. If not, the function is not even.

### Example:
If the function given is \( f(x) = x^2 + 4 \), let's validate if it's even:

- Substitute \( -x \) for \( x \):
\[
f(-x) = (-x)^2 + 4 = x^2 + 4.
\]
- Since \( f(-x) = f(x) = x^2 + 4 \), this confirms the function is even.

This process helps students verify evenness by showing that the function maintains symmetry about the y-axis. For functions that don't satisfy \( f(-x) = f(x) \), students would conclude the function is not even.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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