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.