To find the polynomial equivalent to [tex]\( f(f(x)) \)[/tex] given the function [tex]\( f(x) = -4x + 7 \)[/tex]:
1. First, calculate [tex]\( f(x) \)[/tex]. According to the problem, [tex]\( f(x) = -4x + 7 \)[/tex].
2. Next, apply the function [tex]\( f \)[/tex] to [tex]\( f(x) \)[/tex]. This means we need to find [tex]\( f(f(x)) \)[/tex].
3. Start with the inner function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = -4x + 7
\][/tex]
4. Substitute [tex]\( f(x) \)[/tex] into itself:
[tex]\[
f(f(x)) = f(-4x + 7)
\][/tex]
5. Now, apply the function [tex]\( f \)[/tex] to [tex]\( -4x + 7 \)[/tex]:
[tex]\[
f(-4x + 7) = -4(-4x + 7) + 7
\][/tex]
6. Simplify the expression:
[tex]\[
-4(-4x + 7) + 7 = 16x - 28 + 7
\][/tex]
7. Combine like terms:
[tex]\[
16x - 28 + 7 = 16x - 21
\][/tex]
Thus, the polynomial equivalent to [tex]\( f(f(x)) \)[/tex] is [tex]\( 16x - 21 \)[/tex].
Therefore, the correct answer is:
C. [tex]\( 16x - 21 \)[/tex]
