Question
\[ f(x) = 3x + 4 \quad g(x) = 2x - 3 \] \[ \text{Find: } f \circ g(x) \]
Ask by Harmon Kirk. in South Africa
Mar 08,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f \circ g(x) = 6x - 5 \)
Solution
1. Write the composition of the functions:
\[
f \circ g(x) = f(g(x))
\]
2. Substitute \( g(x) = 2x - 3 \) into \( f(x) \):
\[
f(g(x)) = f(2x - 3)
\]
3. Use the definition of \( f(x) = 3x + 4 \) with \( x \) replaced by \( 2x - 3 \):
\[
f(2x-3) = 3(2x-3) + 4
\]
4. Simplify the expression:
\[
3(2x-3) + 4 = 6x - 9 + 4 = 6x - 5
\]
5. Thus, the composition is:
\[
f \circ g(x) = 6x - 5
\]
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Simplify this solution Extra Insights
To find \( f \circ g(x) \), we will substitute \( g(x) \) into \( f(x) \). First, calculate \( g(x) \): \[ g(x) = 2x - 3 \] Now, substitute \( g(x) \) into \( f(x) = 3x + 4 \): \[ f(g(x)) = f(2x - 3) = 3(2x - 3) + 4 \] Now, distribute the 3: \[ = 6x - 9 + 4 \] Then simplify: \[ = 6x - 5 \] Thus, \( f \circ g(x) = 6x - 5 \).
