To determine if the function [tex]\( f(x) = 6 x^4 + 7 x^2 \)[/tex] is even, odd, or neither, we follow these steps:
1. Definition of Even Function:
A function [tex]\( f(x) \)[/tex] is even if for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex], [tex]\( f(-x) = f(x) \)[/tex].
2. Definition of Odd Function:
A function [tex]\( f(x) \)[/tex] is odd if for all [tex]\( x \)[/tex] in the domain of [tex]\( f \)[/tex], [tex]\( f(-x) = -f(x) \)[/tex].
3. Calculate [tex]\( f(-x) \)[/tex]:
Let's substitute [tex]\( -x \)[/tex] into the function:
[tex]\[
f(-x) = 6 (-x)^4 + 7 (-x)^2
\][/tex]
Simplify the expression:
[tex]\[
f(-x) = 6(-x)^4 + 7(-x)^2
\][/tex]
Since [tex]\((-x)^4 = x^4\)[/tex] and [tex]\((-x)^2 = x^2\)[/tex], we get:
[tex]\[
f(-x) = 6 x^4 + 7 x^2
\][/tex]
4. Compare [tex]\( f(-x) \)[/tex] with [tex]\( f(x) \)[/tex]:
We have:
[tex]\[
f(x) = 6 x^4 + 7 x^2
\][/tex]
[tex]\[
f(-x) = 6 x^4 + 7 x^2
\][/tex]
Since [tex]\( f(-x) = f(x) \)[/tex], the function satisfies the condition for being an even function.
5. Check if the Function is Odd:
If the function were odd, we should have:
[tex]\[
f(-x) = -f(x)
\][/tex]
However:
[tex]\[
f(-x) = 6 x^4 + 7 x^2
\][/tex]
[tex]\[
-f(x) = -(6 x^4 + 7 x^2) = -6 x^4 - 7 x^2
\][/tex]
Clearly, [tex]\( f(-x) \neq -f(x) \)[/tex].
6. Conclusion:
Since [tex]\( f(-x) = f(x) \)[/tex], the function [tex]\( f(x) = 6 x^4 + 7 x^2 \)[/tex] is an even function. It is not odd, nor is it neither.
Thus, the function [tex]\( f(x) = 6 x^4 + 7 x^2 \)[/tex] is Even.
