The functions in Exercises 11-28 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(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = (x +4)/(x-2)

Verified step by step guidance

1. Start by writing the function as an equation with y: \(y = \frac{x + 4}{x - 2}\).
2. To find the inverse, swap x and y: \(x = \frac{y + 4}{y - 2}\).
3. Solve this new equation for y. Begin by multiplying both sides by \((y - 2)\) to eliminate the denominator: \(x(y - 2) = y + 4\).
4. Distribute x on the left side: \(xy - 2x = y + 4\). Then, collect all terms involving y on one side and constants on the other: \(xy - y = 2x + 4\).
5. Factor y out on the left: \(y(x - 1) = 2x + 4\). Finally, solve for y by dividing both sides by \((x - 1)\): \(y = \frac{2x + 4}{x - 1}\). This expression represents \(f^{-1}(x)\).

Key Concepts

One-to-One Functions

A function is one-to-one if each output corresponds to exactly one input, ensuring it has an inverse. This property is crucial because only one-to-one functions have well-defined inverse functions that reverse the original mapping.

Finding the Inverse Function

To find the inverse of a function, swap the roles of x and y in the equation and solve for y. This process reverses the input-output relationship, producing a function that undoes the effect of the original function.

Verification of Inverse Functions

To verify an inverse function, show that composing the function and its inverse in both orders returns the original input: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This confirms the two functions are true inverses of each other.