If f(x) is an even function, which function must also be even? (1) f(x – 2) (2) f(x) + 3 (3) fl(x + 1) (4) f(x + 1) + 3

Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### 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:
\( \boxed{2} \ f(x) + 3 \)
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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