Question

Question
Which functions are even? Select all that apply.
(2 points)
Responses

f left-parenthesis x right-parenthesis equals Root Index 3 Start Root 8 x End Root
Image with alt text: f left-parenthesis x right-parenthesis equals Root Index 3 Start Root 8 x End Root

f left-parenthesis x right-parenthesis equals log subscript 9 baseline x superscript 6 baseline
Image with alt text: f left-parenthesis x right-parenthesis equals log subscript 9 baseline x superscript 6 baseline

f left-parenthesis x right-parenthesis equals Start Fraction 1 over x superscript 8 baseline plus 7 x superscript 6 baseline End Fraction
Image with alt text: f left-parenthesis x right-parenthesis equals Start Fraction 1 over x superscript 8 baseline plus 7 x superscript 6 baseline End Fraction

f left-parenthesis x right-parenthesis equals e superscript x superscript 8 baseline minus x baseline
Image with alt text: f left-parenthesis x right-parenthesis equals e superscript x superscript 8 baseline minus x baseline

f left-parenthesis x right-parenthesis equals Start Absolute Value 8 x End Absolute Value minus 3
Image with alt text: f left-parenthesis x right-parenthesis equals Start Absolute Value 8 x End Absolute Value minus 3

f left-parenthesis x right-parenthesis equals negative 9 x superscript 10 baseline plus 5 x superscript 4 baseline minus 12 x
Image with alt text: f left-parenthesis x right-parenthesis equals negative 9 x superscript 10 baseline plus 5 x superscript 4 baseline minus 12 x
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Answer 1

To determine which functions are even, we need to check if \( f(-x) = f(x) \) for each given function. Here’s the analysis for each option:

1. \( f(x) = \sqrt[3]{8x} \)
- \( f(-x) = \sqrt[3]{8(-x)} = \sqrt[3]{-8x} = -\sqrt[3]{8x} \neq f(x) \)
- **Not even.**

2. \( f(x) = \log_{9}(x^6) \)
- \( f(-x) = \log_{9}((-x)^6) = \log_{9}(x^6) = f(x) \)
- **Even.**

3. \( f(x) = \frac{1}{x^8 + 7x^6} \)
- \( f(-x) = \frac{1}{(-x)^8 + 7(-x)^6} = \frac{1}{x^8 + 7x^6} = f(x) \)
- **Even.**

4. \( f(x) = e^{x^8} - x \)
- \( f(-x) = e^{(-x)^8} - (-x) = e^{x^8} + x \neq f(x) \)
- **Not even.**

5. \( f(x) = |8x| - 3 \)
- \( f(-x) = |8(-x)| - 3 = |8x| - 3 = f(x) \)
- **Even.**

6. \( f(x) = -9x^{10} + 5x^4 - 12x \)
- \( f(-x) = -9(-x)^{10} + 5(-x)^4 - 12(-x) = -9x^{10} + 5x^4 + 12x \neq f(x) \)
- **Not even.**

**Conclusion: The even functions are:**
- \( f(x) = \log_{9}(x^6) \)
- \( f(x) = \frac{1}{x^8 + 7x^6} \)
- \( f(x) = |8x| - 3 \)