find the domain and range of inverse function h(x)=5-9x

To find the domain and range of the inverse function, we first need to determine the inverse of the given function \( h(x) = 5 - 9x \).

### Step 1: Express \( h(x) = 5 - 9x \) as \( y = 5 - 9x \).
\[
y = 5 - 9x
\]

### Step 2: Swap \( x \) and \( y \) to find the inverse.
\[
x = 5 - 9y
\]

### Step 3: Solve for \( y \).
\[
9y = 5 - x
\]
\[
y = \frac{5 - x}{9}
\]
Thus, the inverse function is:
\[
h^{-1}(x) = \frac{5 - x}{9}
\]

### Step 4: Find the domain and range of the inverse function.
1. **Domain of \( h^{-1}(x) \)**: The domain is all the possible values of \( x \) for which the inverse function is defined. Since \( h^{-1}(x) = \frac{5 - x}{9} \) is a linear function, it is defined for all real values of \( x \). Therefore, the domain is:
   \[
   \text{Domain of } h^{-1}(x): (-\infty, \infty)
   \]

2. **Range of \( h^{-1}(x) \)**: The range of the inverse function is the set of output values, which is also all real numbers because the inverse is a linear function. Therefore, the range is:
   \[
   \text{Range of } h^{-1}(x): (-\infty, \infty)
   \]

### Final Answer:
- Domain of \( h^{-1}(x) \): \( (-\infty, \infty) \)
- Range of \( h^{-1}(x) \): \( (-\infty, \infty) \)

Would you like more details, or have any questions?

Here are 5 related questions:
1. What is the process of finding the inverse of a quadratic function?
2. How do you find the domain of a square root function?
3. Can a function have an inverse if it is not one-to-one?
4. How do horizontal line tests determine if a function has an inverse?
5. What are the steps to graphing an inverse function?

**Tip:** When finding the inverse of a function, always check if the original function is one-to-one to ensure the inverse exists.

Explore the step-by-step process to find the domain and range of the inverse function h(x) = 5 - 9x. Learn about inverse functions, linear equations, and how to determine the domain and range of a function. Ideal for students in Grades 9-11.

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