https://file.aiquickdraw.com/mathbot/file/1727193445782qjby5752.jpg

From the image, we have a function \( y = h(x) \), and the graph of \( y = h(x) \) is shown. Let's break down the tasks outlined:

### 1. **Sketching the graph of \( y = h^{-1}(x) \) (the inverse function):**
   To sketch \( y = h^{-1}(x) \), we reflect the graph of \( y = h(x) \) over the line \( y = x \). This means that for each point \( (a, b) \) on \( y = h(x) \), the corresponding point \( (b, a) \) will appear on \( y = h^{-1}(x) \).

### 2. **Creating a table of values:**
   From the graph of \( y = h(x) \), we can pick a few points and then swap the coordinates to get points for \( y = h^{-1}(x) \).
   - If \( h(0) = 1 \), then \( h^{-1}(1) = 0 \).
   - If \( h(1) = 2 \), then \( h^{-1}(2) = 1 \).
   - If \( h(2) = 3 \), then \( h^{-1}(3) = 2 \).

### 3. **Finding the domain and range of both functions:**
   - For \( y = h(x) \):
     - **Domain**: The graph of \( h(x) \) seems to be defined for \( x \in [0, \infty) \), meaning \( x \geq 0 \).
     - **Range**: The values of \( y = h(x) \) seem to range from \( y \in [1, \infty) \), meaning \( y \geq 1 \).
   
   - For \( y = h^{-1}(x) \):
     - **Domain**: The domain of the inverse function is the range of the original function, so \( x \in [1, \infty) \).
     - **Range**: The range of the inverse function is the domain of the original function, so \( y \in [0, \infty) \).

Would you like more details or clarifications?

### 5 Related Questions:
1. How do you find the inverse of a function algebraically?
2. What are the conditions for a function to have an inverse?
3. How do you determine the domain and range of a function from its graph?
4. What is the significance of reflecting over the line \( y = x \) for inverse functions?
5. Can all functions have inverses, and if not, why?

### Tip:
Always verify if the function is one-to-one before determining its inverse. This ensures that each input has exactly one output, which is a necessary condition for the inverse to exist.

The function y = h(x) is entirely defined by the graph provided. Sketch a graph of y = h^{-1}(x), and write the domain and range for both y = h(x) and y = h^{-1}(x).

Learn how to sketch the inverse of a function and determine the domain and range of both the original and inverse functions. This includes graph reflections and domain/range identification.

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