Math Snap

STEP 1

1. We are given a one-to-one function $f(x) = 2x + 3$.
2. We need to find the inverse of this function.
3. We need to graph both the function and its inverse on the same set of axes.

STEP 2

1. Verify the given inverse function.
2. Derive the inverse function from the original function.
3. Graph the original function.
4. Graph the inverse function.
5. Verify that the graphs are reflections over the line $y = x$.

STEP 3

Verify the given inverse function $f^{-1}(x) = \frac{x-3}{2}$.
To verify, substitute $f^{-1}(x)$ into $f(x)$ and check if $f(f^{-1}(x)) = x$:
$$f(f^{-1}(x)) = f\left(\frac{x-3}{2}\right) = 2\left(\frac{x-3}{2}\right) + 3$$ Simplify:
$$= (x-3) + 3 = x$$ The given inverse function is correct.

STEP 4

Derive the inverse function from the original function $f(x) = 2x + 3$.
1. Replace $f(x)$ with $y$: $y = 2x + 3$.
2. Swap $x$ and $y$: $x = 2y + 3$.
3. Solve for $y$:
$$x - 3 = 2y$$ $$y = \frac{x - 3}{2}$$ Thus, the inverse function is $f^{-1}(x) = \frac{x-3}{2}$.

STEP 5

Graph the original function $f(x) = 2x + 3$.
1. Identify the y-intercept: $(0, 3)$.
2. Use the slope $2$ to find another point: From $(0, 3)$, move up 2 units and right 1 unit to $(1, 5)$.
3. Draw the line through these points.

STEP 6

Graph the inverse function $f^{-1}(x) = \frac{x-3}{2}$.
1. Identify the y-intercept: $(0, -\frac{3}{2})$.
2. Use the slope $\frac{1}{2}$ to find another point: From $(0, -\frac{3}{2})$, move up 1 unit and right 2 units to $(2, -\frac{1}{2})$.
3. Draw the line through these points.

Verify that the graphs are reflections over the line $y = x$.
1. Draw the line $y = x$.
2. Check that each point on $f(x)$ corresponds to a reflected point on $f^{-1}(x)$.
The inverse function is:
$$f^{-1}(x) = \frac{x-3}{2}$$