Sure, let's go through this problem step by step.
The given function is [tex]\( q(x) = 3x^4 - 2x^2 \)[/tex].
We need to find [tex]\( q(2x^2) \)[/tex]. This means we will substitute [tex]\( 2x^2 \)[/tex] in place of [tex]\( x \)[/tex] in the original function [tex]\( q(x) \)[/tex].
1. Start with the original function:
[tex]\[
q(x) = 3x^4 - 2x^2
\][/tex]
2. Substitute [tex]\( 2x^2 \)[/tex] for [tex]\( x \)[/tex]:
[tex]\[
q(2x^2) = 3(2x^2)^4 - 2(2x^2)^2
\][/tex]
3. Calculate [tex]\( (2x^2)^4 \)[/tex]:
[tex]\[
(2x^2)^4 = 2^4 \cdot (x^2)^4 = 16x^8
\][/tex]
4. Calculate [tex]\( (2x^2)^2 \)[/tex]:
[tex]\[
(2x^2)^2 = 2^2 \cdot (x^2)^2 = 4x^4
\][/tex]
5. Substitute these values back into the function:
[tex]\[
q(2x^2) = 3(16x^8) - 2(4x^4)
\][/tex]
6. Simplify the expression:
[tex]\[
q(2x^2) = 48x^8 - 8x^4
\][/tex]
So, the function [tex]\( q(x) = 3x^4 - 2x^2 \)[/tex] when substituted with [tex]\( 2x^2 \)[/tex] is:
[tex]\[
q(2x^2) = 48x^8 - 8x^4
\][/tex]
Hence, the result is:
[tex]\[
q(x) = 3x^4 - 2x^2 \quad ; \quad q(2x^2) = 48x^8 - 8x^4
\][/tex]
