Find the inverse of f(x)=(x−10)/(x+10).

Verified step by step guidance

1. Step 1: To find the inverse of a function, we first replace the function notation f(x) with y. So, we rewrite the function as y = (x - 10) / (x + 10).
2. Step 2: The next step in finding the inverse of a function is to swap x and y. This gives us x = (y - 10) / (y + 10).
3. Step 3: Now, we need to solve this equation for y. To do this, we first multiply both sides by (y + 10) to get rid of the denominator on the right side. This gives us x(y + 10) = y - 10.
4. Step 4: Distribute x on the left side to get xy + 10x = y - 10. Then, rearrange the equation to get all terms involving y on one side and constants on the other. This gives us xy - y = -10 - 10x.
5. Step 5: Factor out y on the left side to get y(x - 1) = -10 - 10x. Finally, divide both sides by (x - 1) to solve for y. This gives us y = (-10 - 10x) / (x - 1), which is the inverse of the original function.

Key Concepts

Function Inverses

An inverse function essentially reverses the effect of the original function. If f(x) takes an input x and produces an output y, then the inverse function, denoted as f⁻¹(y), takes y back to x. To find the inverse, one typically swaps the roles of x and y in the equation and solves for y.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of f(x) = (x - 10)/(x + 10), both the numerator and denominator are linear polynomials. Understanding the properties of rational functions, such as their domain and asymptotic behavior, is crucial for analyzing their inverses.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables. This skill is essential when finding inverses, as it often requires moving terms across the equation, factoring, or using operations like addition, subtraction, multiplication, and division to solve for the desired variable.