To find the values of \( f \circ f(x) \) and \( f \circ f(-2) \) when \( f(x) = 8x + 7 \), we must follow these steps:
### Step-by-Step Solution
1. Understand the function composition:
- \( f \circ f(x) \) implies the function \( f \) is applied to the result of \( f(x) \). This is written as \( f(f(x)) \).
2. Calculate \( f(f(x)) \):
- Start with \( f(x) = 8x + 7 \).
- Apply \( f \) to \( f(x) \):
[tex]\[
f(f(x)) = f(8x + 7)
\][/tex]
- To find \( f(8x + 7) \), substitute \( 8x + 7 \) into the function \( f \):
[tex]\[
f(8x + 7) = 8(8x + 7) + 7
\][/tex]
- Simplify the expression:
[tex]\[
f(8x + 7) = 8 \times 8x + 8 \times 7 + 7 = 64x + 56 + 7 = 64x + 63
\][/tex]
- Therefore,
[tex]\[
f(f(x)) = 64x + 63
\][/tex]
3. Calculate \( f(f(-2)) \):
- First, compute \( f(-2) \):
[tex]\[
f(-2) = 8(-2) + 7 = -16 + 7 = -9
\][/tex]
- Next, apply \( f \) to this result:
[tex]\[
f(-9) = 8(-9) + 7 = -72 + 7 = -65
\][/tex]
### Final Results
- The value of \( f \circ f(x) \) is:
[tex]\[
f(f(x)) = 64x + 63
\][/tex]
- The value of \( f \circ f(-2) \) is:
[tex]\[
f(f(-2)) = -65
\][/tex]
Hence, the answers are:
- \( f \circ f(x) = 64x + 63 \)
- [tex]\( f \circ f(-2) = -65 \)[/tex]
