Parry Bolton
09/03/2023 · Elementary School
\( \left. \begin{array} { l } { f ( x ) = x ^ { 2 } - 4 x + \quad g ( x ) = x - 1 \quad h ( x ) = 8 x - 3 \quad r ( x ) = x + 2 } \\ { f ( h ( x ) ) \quad g ( 4 h ( - 6 ) ) } \end{array} \right. \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the problem, we need to evaluate the expressions \( f(h(x)) \) and \( g(4h(-6)) \) using the given functions.
1. **Evaluate \( h(-6) \)**:
\[
h(x) = 8x - 3
\]
Substituting \( x = -6 \):
\[
h(-6) = 8(-6) - 3 = -48 - 3 = -51
\]
2. **Evaluate \( 4h(-6) \)**:
\[
4h(-6) = 4 \times -51 = -204
\]
3. **Evaluate \( g(4h(-6)) \)**:
\[
g(x) = x - 1
\]
Substituting \( x = -204 \):
\[
g(-204) = -204 - 1 = -205
\]
4. **Evaluate \( f(h(x)) \)**:
First, we need to find \( h(x) \):
\[
h(x) = 8x - 3
\]
Now substitute \( h(x) \) into \( f(x) \):
\[
f(x) = x^2 - 4x
\]
So we need to evaluate \( f(h(x)) = f(8x - 3) \):
\[
f(8x - 3) = (8x - 3)^2 - 4(8x - 3)
\]
Now calculate \( (8x - 3)^2 \):
\[
(8x - 3)^2 = 64x^2 - 48x + 9
\]
Now calculate \( -4(8x - 3) \):
\[
-4(8x - 3) = -32x + 12
\]
Combine these results:
\[
f(8x - 3) = (64x^2 - 48x + 9) + (-32x + 12) = 64x^2 - 80x + 21
\]
5. **Final Results**:
- \( g(4h(-6)) = -205 \)
- \( f(h(x)) = 64x^2 - 80x + 21 \)
Thus, the final answers are:
\[
f(h(x)) = 64x^2 - 80x + 21
\]
\[
g(4h(-6)) = -205
\]
Quick Answer
\( f(h(x)) = 64x^2 - 80x + 21 \)
\( g(4h(-6)) = -205 \)
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