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If [tex]f(x) = 8x + 7[/tex], then find the value of [tex]f \circ f(x)[/tex] and [tex]f \circ f(-2)[/tex].

Sagot :

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]