1 We have the function f(x) = (1/2)x+1.

a Find the inverse function of f(x) and call it g(x).

b Draw both functions in a coordinate system together with the line y = x.

C Use the graph to explain that f(x) and g(x) are inverse functions.

### Part (a): Finding the inverse function of \( f(x) = \frac{1}{2}x + 1 \)

To find the inverse function, \( g(x) \), we follow these steps:

1. Start with the equation for \( f(x) \):
   \[
   y = \frac{1}{2}x + 1
   \]

2. Swap \( x \) and \( y \), since the inverse function reverses the roles of \( x \) and \( y \):
   \[
   x = \frac{1}{2}y + 1
   \]

3. Solve for \( y \):
   \[
   x - 1 = \frac{1}{2}y
   \]
   \[
   y = 2(x - 1)
   \]
   \[
   y = 2x - 2
   \]

Thus, the inverse function \( g(x) \) is:
\[
g(x) = 2x - 2
\]

### Part (b): Drawing both functions and the line \( y = x \)

To plot the functions \( f(x) = \frac{1}{2}x + 1 \) and \( g(x) = 2x - 2 \) along with the line \( y = x \), we need to sketch these graphs:

1. **\( f(x) = \frac{1}{2}x + 1 \)**: This is a straight line with slope \( \frac{1}{2} \) and y-intercept at \( (0, 1) \).
   
2. **\( g(x) = 2x - 2 \)**: This is another straight line with slope \( 2 \) and y-intercept at \( (0, -2) \).

3. **Line \( y = x \)**: This is the identity line that bisects the coordinate plane, reflecting all points symmetrically.

These three lines should intersect at points where they reflect each other. I can create the graph for you if you upload a coordinate system or graphing tool.

### Part (c): Explaining the Inverse Property Using the Graph

The graph visually shows that the functions \( f(x) \) and \( g(x) \) are **mirror images** of each other with respect to the line \( y = x \). Here’s why this confirms they are inverse functions:

- For any point \( (a, b) \) on the graph of \( f(x) \), the point \( (b, a) \) lies on the graph of \( g(x) \).
- This symmetry about the line \( y = x \) means that \( g(f(x)) = x \) and \( f(g(x)) = x \), which satisfies the definition of inverse functions.

Would you like to see a graph of these functions? I can also help with further details or specific clarifications.

### Relative Questions:
1. What is the domain and range of the inverse function \( g(x) = 2x - 2 \)?
2. How would the graph change if \( f(x) \) had a different slope?
3. Can you verify that \( f(g(x)) = x \) algebraically?
4. How does the y-intercept of \( f(x) \) relate to the x-intercept of \( g(x) \)?
5. What happens to the inverse function when \( f(x) \) is not a linear function?

### Tip:
When finding the inverse of a linear function, remember that the slope of the inverse is the reciprocal of the original function's slope.

We have the function f(x) = (1/2)x + 1.

a) Find the inverse function of f(x) and call it g(x).

b) Draw both functions in a coordinate system together with the line y = x.

c) Use the graph to explain that f(x) and g(x) are inverse functions.

This tutorial shows how to find the inverse of the linear function f(x) = (1/2)x + 1, resulting in g(x) = 2x - 2. The solution explains the relationship between the two functions using a graph and the line y = x, suitable for Grades 9-11.

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