For $$ f(x)=2 x-1 $$ and $$ g(x)=\frac{1}{2}(x+1) $$, find $$ (f \circ g)(x) $$ and $$ (g \circ f)(x) $$. Then determine whether $$ (f \circ g)(x)=(g \circ f)(x) $$. What is $$ (f \circ g)(x) $$ ? $$ (f \circ g)(x)=x $$ What is $$ (g \circ f)(x) $$ ? $$ (g \circ f)(x)=\square $$

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UpStudy AI · Accepted Answer

Let's solve the problem step by step.

### Step 1. Compute $$ (f \circ g)(x) $$

By definition,  
$$
(f \circ g)(x) = f(g(x))
$$

Given:
$$
f(x) = 2x-1 \quad \text{and} \quad g(x) = \frac{1}{2}(x+1)
$$

Substitute $$ g(x) $$ into $$ f(x) $$:
$$
f(g(x)) = 2\left(\frac{1}{2}(x+1)\right) - 1
$$

Simplify the expression:
$$
2\left(\frac{1}{2}(x+1)\right) = x+1
$$
$$
x+1 - 1 = x
$$

Thus:
$$
(f \circ g)(x) = x
$$

### Step 2. Compute $$ (g \circ f)(x) $$

By definition,  
$$
(g \circ f)(x) = g(f(x))
$$

Substitute $$ f(x) $$ into $$ g(x) $$:
$$
g(f(x)) = \frac{1}{2}\left((2x-1)+1\right)
$$

Simplify the expression within the parentheses:
$$
(2x-1)+1 = 2x
$$

Now compute:
$$
\frac{1}{2}(2x) = x
$$

Thus:
$$
(g \circ f)(x) = x
$$

### Step 3. Compare the Two Compositions

We found:
$$
(f \circ g)(x) = x \quad \text{and} \quad (g \circ f)(x) = x
$$

Therefore, they are equal:
$$
(f \circ g)(x) = (g \circ f)(x)
$$

### Final Answer

- $$ (f \circ g)(x) = x $$
- $$ (g \circ f)(x) = x $$

Since both compositions result in $$ x $$, they are indeed equal.
$$ (f \circ g)(x) = x $$ and $$ (g \circ f)(x) = x $$. Therefore, $$ (f \circ g)(x) = (g \circ f)(x) $$.

For \( f(x)=2 x-1 \) and \( g(x)=\frac{1}{2}(x+1) \), find \( (f \circ g)(x) \) and \( (g \circ f)(x) \). Then determine whether \( (f \circ g)(x)=(g \circ f)(x) \). What is \( (f \circ g)(x) \) ? \( (f \circ g)(x)=x \) What is \( (g \circ f)(x) \) ? \( (g \circ f)(x)=\square \)

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