Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1

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Start by writing the function explicitly: \(f(x) = x^{2} - 2x\) with the domain \(x \leq 1\).

To find the inverse function \(f^{-1}(x)\), first replace \(f(x)\) with \(y\): \(y = x^{2} - 2x\).

Next, solve the equation \(y = x^{2} - 2x\) for \(x\) in terms of \(y\). Rearrange it as a quadratic equation: \(x^{2} - 2x - y = 0\).

Use the quadratic formula to solve for \(x\): \(x = \frac{2 \pm \sqrt{(-2)^{2} - 4 \cdot 1 \cdot (-y)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4y}}{2} = 1 \pm \sqrt{1 + y}\).

Since the original domain is \(x \leq 1\), choose the branch of the solution that satisfies this domain (which will be \(x = 1 - \sqrt{1 + y}\)). Then express the inverse function as \(f^{-1}(x) = 1 - \sqrt{1 + x}\). Finally, determine the domain and range of \(f^{-1}\) by considering the range and domain of \(f\), respectively.

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주요 개념

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Inverse Functions

An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x within their domains. Finding the inverse involves solving y = f(x) for x in terms of y.

Domain and Range Restrictions

To ensure a function has an inverse, it must be one-to-one, often requiring domain restrictions. The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹. Identifying these sets is crucial for correctly defining and verifying the inverse.

Verification of Inverse Functions

Verifying an inverse involves composing the function and its inverse in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions should simplify to x within the appropriate domains. This confirms the correctness of the inverse function and the domain-range assignments.

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.f(x) = x² − 2x, x ≤ 1