For $$ f(x)=6 x $$ and $$ g(x)=x+2 $$, find the following functions. a. $$ (f \circ g)(x) ; b .(g \circ f)(x) ; $$ c. $$ (f \circ g)(2) ; $$ d. $$ (g \circ f)(2) $$

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UpStudy AI · Accepted Answer

Let's find each of the requested compositions step by step.

Given:
- $$ f(x) = 6x $$
- $$ g(x) = x + 2 $$

### a. $$ (f \circ g)(x) $$

**Definition:**
$$ (f \circ g)(x) = f(g(x)) $$

**Calculation:**
$$
\begin{align*}
(f \circ g)(x) &= f(g(x)) \
&= f(x + 2) \
&= 6(x + 2) \
&= 6x + 12
\end{align*}
$$

**Answer:**
$$ (f \circ g)(x) = 6x + 12 $$

---

### b. $$ (g \circ f)(x) $$

**Definition:**
$$ (g \circ f)(x) = g(f(x)) $$

**Calculation:**
$$
\begin{align*}
(g \circ f)(x) &= g(f(x)) \
&= g(6x) \
&= 6x + 2
\end{align*}
$$

**Answer:**
$$ (g \circ f)(x) = 6x + 2 $$

---

### c. $$ (f \circ g)(2) $$

**Calculation:**
$$
\begin{align*}
(f \circ g)(2) &= f(g(2)) \
&= f(2 + 2) \
&= f(4) \
&= 6 \times 4 \
&= 24
\end{align*}
$$

**Answer:**
$$ (f \circ g)(2) = 24 $$

---

### d. $$ (g \circ f)(2) $$

**Calculation:**
$$
\begin{align*}
(g \circ f)(2) &= g(f(2)) \
&= g(6 \times 2) \
&= g(12) \
&= 12 + 2 \
&= 14
\end{align*}
$$

**Answer:**
$$ (g \circ f)(2) = 14 $$

---

### Summary of Results

a. $$ (f \circ g)(x) = 6x + 12 $$

b. $$ (g \circ f)(x) = 6x + 2 $$

c. $$ (f \circ g)(2) = 24 $$

d. $$ (g \circ f)(2) = 14 $$
a. $$ (f \circ g)(x) = 6x + 12 $$ b. $$ (g \circ f)(x) = 6x + 2 $$ c. $$ (f \circ g)(2) = 24 $$ d. $$ (g \circ f)(2) = 14 $$

For \( f(x)=6 x \) and \( g(x)=x+2 \), find the following functions. a. \( (f \circ g)(x) ; b .(g \circ f)(x) ; \) c. \( (f \circ g)(2) ; \) d. \( (g \circ f)(2) \)

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