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 \( g(x) = 13x - 7 \):
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Domain: The domain of \( g(x) \) is all real numbers \( (-\infty, \infty) \) because the function is a linear function with no restrictions.
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Range: The range of a linear function that continues indefinitely in both directions is also all real numbers \( (-\infty, \infty) \).
For the inverse function \( g^{-1}(x) = \frac{x + 7}{13} \):
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Domain: The domain of \( g^{-1}(x) \) is also all real numbers \( (-\infty, \infty) \).
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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 \( g^{-1}(x) \): \( (-\infty, \infty) \)