Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Transcribed Image Text:### Question 7: Even Functions
If \( f(x) \) is an even function, which function must also be even?
Options:
1. \( f(x - 2) \)
2. \( f(x) + 3 \)
3. \( f(x + 1) \)
4. \( f(x + 1) + 3 \)
**Explanation:**
An even function \( f(x) \) is defined such that \( f(x) = f(-x) \) for all \( x \) in the domain of \( f \). This property can help us determine which modifications to \( f(x) \) preserve its even nature.
1. **\( f(x - 2) \)**:
- To check if \( f(x - 2) \) is even, we substitute \( -x \) and compare:
\( f(x - 2) \) and \( f(-x - 2) \).
Since \( x - 2 \neq -x - 2 \), \( f(x - 2) \) is not necessarily even.
2. **\( f(x) + 3 \)**:
- Adding a constant to \( f(x) \) does not change the symmetry of \( f(x) \).
Therefore, if \( f(x) \) is even, \( f(x) + 3 \) remains even.
3. **\( f(x + 1) \)**:
- To check if \( f(x + 1) \) is even, we substitute \( -x \) and compare:
\( f(x + 1) \) and \( f(-x + 1) \).
Since \( x + 1 \neq -x + 1 \), \( f(x + 1) \) is not necessarily even.
4. **\( f(x + 1) + 3 \)**:
- Similar to option 3, since \( f(x + 1) \) is not necessarily even, adding a constant does not change this.
Therefore, \( f(x + 1) + 3 \) is not necessarily even.
**Conclusion:**
The function that must also be even if \( f(x) \) is an even function is:
\( \boxed{2} \ f(x) + 3 \)
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