Question
\[ f(x) = 3x + 4 \quad g(x) = 2x - 3 \] \[ \text{Find: } f \circ g(x) \]
Ask by Martinez Williams. in the United States
Jan 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f \circ g(x) = 6x - 5 \)
Solution
To find the composition \( f \circ g(x) \), we need to apply the function \( f \) to the result of \( g(x) \). Here's a step-by-step breakdown:
1. **Given Functions:**
\[
f(x) = 3x + 4 \quad \text{and} \quad g(x) = 2x - 3
\]
2. **Composition \( f \circ g(x) \):**
\[
f \circ g(x) = f(g(x))
\]
3. **Substitute \( g(x) \) into \( f \):**
\[
f(g(x)) = f(2x - 3)
\]
4. **Apply the function \( f \) to \( 2x - 3 \):**
\[
f(2x - 3) = 3(2x - 3) + 4
\]
5. **Simplify the expression:**
\[
= 3 \times 2x - 3 \times 3 + 4
\]
\[
= 6x - 9 + 4
\]
\[
= 6x - 5
\]
6. **Final Result:**
\[
f \circ g(x) = 6x - 5
\]
**Therefore, the composition of \( f \) and \( g \) is:**
\[
f \circ g(x) = 6x - 5
\]
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Simplify this solution The Deep Dive
To find \( f \circ g(x) \), we need to substitute \( g(x) \) into \( f(x) \). First, calculate \( g(x) \): \[ g(x) = 2x - 3 \] Now, substitute \( g(x) \) into \( f(x) \): \[ f(g(x)) = f(2x - 3) = 3(2x - 3) + 4 \] Distributing \( 3 \): \[ = 6x - 9 + 4 \] Now combine like terms: \[ = 6x - 5 \] Thus, \( f \circ g(x) = 6x - 5 \).
