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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주요 개념

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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.