Answer :
To determine which of the functions is an even function, we need to understand what an even function is. An even function is one where [tex]\( f(x) = f(-x) \)[/tex] for every value of [tex]\( x \)[/tex]. This means the function is symmetric about the y-axis.
Let's evaluate each function step-by-step:
1. Function [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 5(-x)^6 - 7(-x)^4 + 13
\][/tex]
- Simplify:
[tex]\[
= 5x^6 - 7x^4 + 13
\][/tex]
- Since [tex]\( f(-x) = f(x) \)[/tex], this function is even.
2. Function [tex]\( f(x) = 12x^8 - 4x^6 + x \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 12(-x)^8 - 4(-x)^6 + (-x)
\][/tex]
- Simplify:
[tex]\[
= 12x^8 - 4x^6 - x
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
3. Function [tex]\( f(x) = 8x^6 - 4x^3 + 10 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 8(-x)^6 - 4(-x)^3 + 10
\][/tex]
- Simplify:
[tex]\[
= 8x^6 + 4x^3 + 10
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
4. Function [tex]\( f(x) = \frac{6x^3 + 8x}{x+2} \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = \frac{6(-x)^3 + 8(-x)}{-x + 2}
\][/tex]
- Simplify:
[tex]\[
= \frac{-6x^3 - 8x}{-x + 2}
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
In conclusion, the only even function from the list provided is:
[tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex].
Let's evaluate each function step-by-step:
1. Function [tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 5(-x)^6 - 7(-x)^4 + 13
\][/tex]
- Simplify:
[tex]\[
= 5x^6 - 7x^4 + 13
\][/tex]
- Since [tex]\( f(-x) = f(x) \)[/tex], this function is even.
2. Function [tex]\( f(x) = 12x^8 - 4x^6 + x \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 12(-x)^8 - 4(-x)^6 + (-x)
\][/tex]
- Simplify:
[tex]\[
= 12x^8 - 4x^6 - x
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
3. Function [tex]\( f(x) = 8x^6 - 4x^3 + 10 \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = 8(-x)^6 - 4(-x)^3 + 10
\][/tex]
- Simplify:
[tex]\[
= 8x^6 + 4x^3 + 10
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
4. Function [tex]\( f(x) = \frac{6x^3 + 8x}{x+2} \)[/tex]:
- Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
f(-x) = \frac{6(-x)^3 + 8(-x)}{-x + 2}
\][/tex]
- Simplify:
[tex]\[
= \frac{-6x^3 - 8x}{-x + 2}
\][/tex]
- Since [tex]\( f(-x) \neq f(x) \)[/tex], this function is not even.
In conclusion, the only even function from the list provided is:
[tex]\( f(x) = 5x^6 - 7x^4 + 13 \)[/tex].
