To determine whether the function [tex]\( f(x) = 3x^2 \)[/tex] is even or odd, we will use the definitions of even and odd functions.
1. Evaluating [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 3(-x)^2
\][/tex]
Since [tex]\((-x)^2 = x^2\)[/tex], we have:
[tex]\[
f(-x) = 3x^2
\][/tex]
2. Checking the conditions for even and odd functions:
- A function is even if [tex]\( f(-x) = f(x) \)[/tex].
- A function is odd if [tex]\( f(-x) = -f(x) \)[/tex].
3. Comparing [tex]\( f(-x) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 3x^2
\][/tex]
[tex]\[
f(-x) = 3x^2
\][/tex]
Since [tex]\( f(-x) = 3x^2 \)[/tex] and [tex]\( f(x) = 3x^2 \)[/tex], we have:
[tex]\[
f(-x) = f(x)
\][/tex]
Thus, the function [tex]\( f(x) = 3x^2 \)[/tex] satisfies the condition for being an even function.
Conclusion:
- The function [tex]\( f(x) = 3x^2 \)[/tex] is even because [tex]\( f(-x) = f(x) \)[/tex].
So, the correct statement is:
- The function is even because [tex]\( f(-x) = f(x) \)[/tex].
