Question

Find the inverse of g(x)=13x−7
and then find its domain and range.

Answer 1

To find the inverse of the function \( g(x) = 13x - 7 \), follow these steps:

### Step 1: Replace \( g(x) \) with \( y \)
Let \( y = g(x) = 13x - 7 \).

### Step 2: Solve for \( x \)
We need to solve for \( x \) in terms of \( y \):

\[
y = 13x - 7
\]

Add 7 to both sides:

\[
y + 7 = 13x
\]

Now, divide both sides by 13:

\[
x = \frac{y + 7}{13}
\]

### Step 3: Write the inverse function
Now we switch \( x \) and \( y \) to express the inverse function:

\[
g^{-1}(x) = \frac{x + 7}{13}
\]

### Step 4: Determine the domain and range
For the function \( g(x) = 13x - 7 \):

- **Domain**: The domain of \( g(x) \) is all real numbers \( (-\infty, \infty) \) because the function is a linear function with no restrictions.

- **Range**: The range of a linear function that continues indefinitely in both directions is also all real numbers \( (-\infty, \infty) \).

For the inverse function \( g^{-1}(x) = \frac{x + 7}{13} \):

- **Domain**: The domain of \( g^{-1}(x) \) is also all real numbers \( (-\infty, \infty) \).

- **Range**: The range is also all real numbers \( (-\infty, \infty) \).

### Summary
The inverse function is

\[
g^{-1}(x) = \frac{x + 7}{13}
\]

- **Domain of \( g(x) \)**: \( (-\infty, \infty) \)
- **Range of \( g(x) \)**: \( (-\infty, \infty) \)
- **Domain of \( g^{-1}(x) \)**: \( (-\infty, \infty) \)
- **Range of \( g^{-1}(x) \)**: \( (-\infty, \infty) \)