Question
Write the composite function \( f(g(x)) \) if \( \begin{array}{ll}\text { a. } f(x)=x^{2} & \text { and } g(x)=2 x+1 \\ \text { b. } f(x)=x+1 & \text { and } g(x)=x^{3}+\sin x .\end{array} \)
Ask by Estrada Barrett. in Nepal
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
a. \( f(g(x)) = (2x + 1)^2 = 4x^{2} + 4x + 1 \)
b. \( f(g(x)) = x^{3} + \sin x + 1 \)
Solution
To find the composite function \( f(g(x)) \), you substitute the function \( g(x) \) into the function \( f(x) \). Let's solve each part step by step.
### a. \( f(x) = x^{2} \) and \( g(x) = 2x + 1 \)
**Composite Function \( f(g(x)) \):**
\[
f(g(x)) = f(2x + 1) = (2x + 1)^{2}
\]
**Simplifying (optional):**
\[
(2x + 1)^{2} = 4x^{2} + 4x + 1
\]
### b. \( f(x) = x + 1 \) and \( g(x) = x^{3} + \sin x \)
**Composite Function \( f(g(x)) \):**
\[
f(g(x)) = f(x^{3} + \sin x) = (x^{3} + \sin x) + 1 = x^{3} + \sin x + 1
\]
### Summary
- **a.** \( f(g(x)) = (2x + 1)^2 = 4x^{2} + 4x + 1 \)
- **b.** \( f(g(x)) = x^{3} + \sin x + 1 \)
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Simplify this solution Bonus Knowledge
To find the composite functions \( f(g(x)) \), we will substitute \( g(x) \) into \( f(x) \). a. Given \( f(x) = x^2 \) and \( g(x) = 2x + 1 \): \[ f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1 \] So, \( f(g(x)) = 4x^2 + 4x + 1 \). b. Given \( f(x) = x + 1 \) and \( g(x) = x^3 + \sin x \): \[ f(g(x)) = f(x^3 + \sin x) = (x^3 + \sin x) + 1 = x^3 + \sin x + 1 \] Thus, \( f(g(x)) = x^3 + \sin x + 1 \).
