Write the composite function $$ f(g(x)) $$ if $$ \begin{array}{ll} ext { a. } f(x)=x^{2} & ext { and } g(x)=2 x+1 \ ext { b. } f(x)=x+1 & ext { and } g(x)=x^{3}+\sin x .\end{array} $$

Question

Write the composite function $$ f(g(x)) $$ if $$ \begin{array}{ll}	ext { a. } f(x)=x^{2} & 	ext { and } g(x)=2 x+1 \ 	ext { b. } f(x)=x+1 & 	ext { and } g(x)=x^{3}+\sin x .\end{array} $$

UpStudy AI · Accepted Answer

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.** $$ f(g(x)) = x^{3} + \sin x + 1 $$
a. $$ f(g(x)) = (2x + 1)^2 = 4x^{2} + 4x + 1 $$ b. $$ f(g(x)) = x^{3} + \sin x + 1 $$

Write the composite function \( f(g(x)) \) if \( \begin{array}{ll}\text { a. } f(x)=x^{2} & \text { and } g(x)=2 x+1 \\ \text { b. } f(x)=x+1 & \text { and } g(x)=x^{3}+\sin x .\end{array} \)

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