The functions in Exercises 93–95 are all one-to-one. For each function, (a) find an equation for f^(-1)x, the inverse function. (b) Verify that your equation is correct by showing that f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. f(x) = (x - 7)/(x + 2)

Verified step by step guidance

1. Step 1: To find the inverse function, f^(-1)(x), start by replacing f(x) with y. So, y = (x - 7) / (x + 2).
2. Step 2: Swap x and y in the equation to begin solving for y. This gives x = (y - 7) / (y + 2).
3. Step 3: Solve for y by cross-multiplying to eliminate the fraction: x(y + 2) = y - 7.
4. Step 4: Rearrange the equation to isolate y on one side. This involves distributing x on the left side, then moving all terms involving y to one side of the equation and constant terms to the other side.
5. Step 5: Verify the inverse by substituting f^(-1)(x) into the original function and checking if f(f^(-1)(x)) = x and substituting f(x) into the inverse function to check if f^(-1)(f(x)) = x.

Key Concepts

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^(-1)(y) takes y back to x. For a function to have an inverse, it must be one-to-one, meaning each output is produced by exactly one input.

Verification of Inverse Functions

To verify that two functions are inverses, we must show that applying one function to the result of the other returns the original input. This is done by proving f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. This step ensures that the functions truly reverse each other's operations.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function f(x) = (x - 7)/(x + 2), the numerator and denominator are both linear polynomials. Understanding the properties of rational functions, including their domains and asymptotic behavior, is essential for finding and verifying their inverses.