Let's find the composition of the function [tex]\( f(x) = -4x + 7 \)[/tex] with itself, denoted as [tex]\( f(f(x)) \)[/tex].
### Step 1: Define the inner function:
The inner function here is [tex]\( f(x) \)[/tex], which is given as:
[tex]\[ f(x) = -4x + 7 \][/tex]
### Step 2: Substitute [tex]\( f(x) \)[/tex] into [tex]\( f \)[/tex]:
We need to find [tex]\( f(f(x)) \)[/tex], which means we substitute [tex]\( f(x) \)[/tex] into itself:
[tex]\[ f(f(x)) = f(-4x + 7) \][/tex]
### Step 3: Apply the function definition:
Substitute [tex]\( -4x + 7 \)[/tex] (the inner function) into the general definition of [tex]\( f(x) \)[/tex]:
[tex]\[ f(-4x + 7) = -4(-4x + 7) + 7 \][/tex]
### Step 4: Simplify the expression inside the function:
Perform the multiplication and addition:
[tex]\[ -4(-4x + 7) = -4 \cdot (-4x) + (-4) \cdot 7 = 16x - 28 \][/tex]
Add the constant 7:
[tex]\[ 16x - 28 + 7 = 16x - 21 \][/tex]
Thus, we have:
[tex]\[ f(f(x)) = 16x - 21 \][/tex]
### Step 5: Compare with the given options:
- A. [tex]\( 16x - 28 \)[/tex]
- B. [tex]\( 16x^2 - 56x + 49 \)[/tex]
- C. [tex]\( 16x - 21 \)[/tex]
- D. [tex]\( 16x^2 - 56x + 56 \)[/tex]
The correct polynomial that matches [tex]\( f(f(x)) = 16x - 21 \)[/tex] is option C.
### Conclusion:
The polynomial equivalent to [tex]\( f(f(x)) \)[/tex] is:
[tex]\[ \boxed{C. \, 16x - 21} \][/tex]
