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Answer :
To determine which function is an even function, we need to check if each function satisfies the property of an even function. A function [tex]\( f(x) \)[/tex] is even if for every [tex]\( x \)[/tex] in the function's domain, [tex]\( f(-x) = f(x) \)[/tex].
Let's analyze each function:
1. [tex]\( f(x) = 12x^8 - 4x^6 + x \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 12(-x)^8 - 4(-x)^6 + (-x) = 12x^8 - 4x^6 - x \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 12x^8 - 4x^6 + x \neq 12x^8 - 4x^6 - x \][/tex]
- Therefore, this function is not even.
2. [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 5(-x)^6 - 7(-x)^4 + 13 = 5x^6 - 7x^4 + 13 \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 5x^6 - 7x^4 + 13 = 5x^6 - 7x^4 + 13 \][/tex]
- This shows the function is even because the expressions are equal for all [tex]\( x \)[/tex].
3. [tex]\( f(x) = 8x^6 - 4x^3 + 10 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 8(-x)^6 - 4(-x)^3 + 10 = 8x^6 + 4x^3 + 10 \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 8x^6 - 4x^3 + 10 \neq 8x^6 + 4x^3 + 10 \][/tex]
- Therefore, this function is not even.
4. [tex]\( f(x) = \frac{6x^3 + 8x}{x + 2} \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = \frac{6(-x)^3 + 8(-x)}{-x + 2} = \frac{-6x^3 - 8x}{-x + 2} \][/tex]
- Comparing [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex] directly may not yield the same result due to the change in signs and terms.
- Subtle changes in the numerator and denominator mean it doesn't appear to be even after simplification.
After the step-by-step examination, the function [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex] is identified to be an even function.
Let's analyze each function:
1. [tex]\( f(x) = 12x^8 - 4x^6 + x \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 12(-x)^8 - 4(-x)^6 + (-x) = 12x^8 - 4x^6 - x \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 12x^8 - 4x^6 + x \neq 12x^8 - 4x^6 - x \][/tex]
- Therefore, this function is not even.
2. [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 5(-x)^6 - 7(-x)^4 + 13 = 5x^6 - 7x^4 + 13 \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 5x^6 - 7x^4 + 13 = 5x^6 - 7x^4 + 13 \][/tex]
- This shows the function is even because the expressions are equal for all [tex]\( x \)[/tex].
3. [tex]\( f(x) = 8x^6 - 4x^3 + 10 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = 8(-x)^6 - 4(-x)^3 + 10 = 8x^6 + 4x^3 + 10 \][/tex]
- Compare [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex]:
[tex]\[ 8x^6 - 4x^3 + 10 \neq 8x^6 + 4x^3 + 10 \][/tex]
- Therefore, this function is not even.
4. [tex]\( f(x) = \frac{6x^3 + 8x}{x + 2} \)[/tex]:
- Substitute [tex]\(-x\)[/tex] for [tex]\(x\)[/tex]:
[tex]\[ f(-x) = \frac{6(-x)^3 + 8(-x)}{-x + 2} = \frac{-6x^3 - 8x}{-x + 2} \][/tex]
- Comparing [tex]\( f(x) \)[/tex] and [tex]\( f(-x) \)[/tex] directly may not yield the same result due to the change in signs and terms.
- Subtle changes in the numerator and denominator mean it doesn't appear to be even after simplification.
After the step-by-step examination, the function [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex] is identified to be an even function.
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