Answer to Question #204137 in Calculus for Jen

Question

1. Create a one-to-one function. It is very important that your function is one-to-one because you will find its inverse. Can't use the function y = x as your one-to-one function.
2. Post a picture of your function's graph.
3. State the window used to graph your function. Carefully choose your window so that it shows as much of the function as possible. Show window on your graphing calculator.
4. Find the inverse of your function.

Answer 1

1. $f(x)=\dfrac{x-2}{2x-5}$

$Domain: (-\infin, \dfrac{5}{2})\cup (\dfrac{5}{2}, \infin)$

 Determine if the function has an inverse. Is the function a one-­to-­one function?

This function passes the Horizontal Line Test which means it is a one-­to-­one function that has an inverse. 

2.

3.

$Domain: (-\infin, \dfrac{5}{2})\cup (\dfrac{5}{2}, \infin)$

$Range: (-\infin, \dfrac{1}{2})\cup (\dfrac{1}{2}, \infin)$

4.

$$f(x)=\dfrac{x-2}{2x-5}$$

Change $f(x)$ to $y$

$$y=\dfrac{x-2}{2x-5}$$

Switch $x$ and $y$

$$x=\dfrac{y-2}{2y-5}$$

Solve for $y$

$$2xy-5x=y-2$$

$$y=\dfrac{5x-2}{2x-1}$$

Change $y$ back to $f^{-1}(x)$

$$f^{-1}(x)=\dfrac{5x-2}{2x-1}$$

$Domain: (-\infin, \dfrac{1}{2})\cup (\dfrac{1}{2}, \infin)$

$Range: (-\infin, \dfrac{5}{2})\cup (\dfrac{5}{2}, \infin)$