Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

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Step 1: To find the inverse of the function \( f(x) = 2x \), start by replacing \( f(x) \) with \( y \), so we have \( y = 2x \).

Step 2: Swap \( x \) and \( y \) to find the inverse function. This gives us \( x = 2y \).

Step 3: Solve for \( y \) in terms of \( x \). Divide both sides by 2 to get \( y = \frac{x}{2} \).

Step 4: Replace \( y \) with \( f^{-1}(x) \) to express the inverse function: \( f^{-1}(x) = \frac{x}{2} \).

Step 5: Verify the inverse by checking \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). Substitute \( f^{-1}(x) \) into \( f(x) \) and vice versa, and simplify to confirm both equal \( x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x), its inverse f⁻¹(x) satisfies the condition that f(f⁻¹(x)) = x for all x in the domain of f⁻¹, and f⁻¹(f(x)) = x for all x in the domain of f. This means that applying the function and then its inverse returns the original input.

Function Composition

Function composition involves combining two functions to create a new function. If you have two functions f and g, the composition f(g(x)) means you apply g first and then apply f to the result. Understanding composition is crucial for verifying the properties of inverse functions, as it demonstrates how they interact with each other.

Linear Functions

A linear function is a polynomial function of degree one, typically expressed in the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the case of f(x) = 2x, it is a linear function with a slope of 2 and no y-intercept. The simplicity of linear functions makes finding their inverses straightforward, as they are one-to-one and can be easily manipulated algebraically.