3. What is the inverse of $$ y=2 x $$

Question

UpStudy AI · Accepted Answer

To find the inverse of the function $$ y = 2x $$, follow these steps:

### Step 1: Replace $$ y $$ with $$ f(x) $$
$$ f(x) = 2x $$

### Step 2: Swap $$ x $$ and $$ y $$
$$ x = 2y $$

### Step 3: Solve for $$ y $$
$$ y = \frac{x}{2} $$

### Step 4: Write the Inverse Function
$$ f^{-1}(x) = \frac{x}{2} $$

### Verification
To ensure that $$ f^{-1}(x) $$ is indeed the inverse of $$ f(x) $$, compose the two functions:

1. **$$ f(f^{-1}(x)) $$:**
   $$
   f\left(\frac{x}{2}\right) = 2 \left(\frac{x}{2}\right) = x
   $$

2. **$$ f^{-1}(f(x)) $$:**
   $$
   f^{-1}(2x) = \frac{2x}{2} = x
   $$

Since both compositions result in $$ x $$, $$ f^{-1}(x) = \frac{x}{2} $$ is indeed the inverse of $$ f(x) = 2x $$.

### Final Answer
The inverse function is:
$$
f^{-1}(x) = \frac{x}{2}
$$
The inverse of $$ y = 2x $$ is $$ f^{-1}(x) = \frac{x}{2} $$.

3. What is the inverse of \( y=2 x \)