To solve the problem, we need to compute the composition of the function [tex]\( f \)[/tex] with itself, i.e., [tex]\( f(f(x)) \)[/tex]. The given function is:
[tex]\[ f(x) = -4x + 7 \][/tex]
First, we substitute [tex]\( f(x) \)[/tex] into the function itself:
[tex]\[ f(f(x)) = f(-4x + 7) \][/tex]
Next, we evaluate [tex]\( f(-4x + 7) \)[/tex] by substituting [tex]\( -4x + 7 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[ f(-4x + 7) = -4(-4x + 7) + 7 \][/tex]
Now, we need to simplify the expression:
[tex]\[ f(-4x + 7) = -4(-4x) + (-4)(7) + 7 \][/tex]
[tex]\[ f(-4x + 7) = 16x - 28 + 7 \][/tex]
[tex]\[ f(-4x + 7) = 16x - 21 \][/tex]
Therefore, the polynomial equivalent to [tex]\( f(f(x)) \)[/tex] is:
[tex]\[ f(f(x)) = 16x - 21 \][/tex]
Among the given options, the correct answer is:
C. [tex]\( 16x - 21 \)[/tex]
