In Exercises 41–44:
a. Find f⁻¹(x).

41. f(x) = 2x + 3, a = −1

검증된 단계별 안내

Start with the given function: \(f(x) = 2x + 3\).

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = 2x + 3\).

Swap the variables \(x\) and \(y\) to get \(x = 2y + 3\); this step reflects the idea that the inverse function reverses the roles of inputs and outputs.

Solve the equation \(x = 2y + 3\) for \(y\): subtract 3 from both sides to get \(x - 3 = 2y\), then divide both sides by 2 to isolate \(y\): \(y = \frac{x - 3}{2}\).

Rewrite \(y\) as \(f^{-1}(x)\) to express the inverse function: \(f^{-1}(x) = \frac{x - 3}{2}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. Finding the inverse involves solving the equation y = f(x) for x in terms of y.

Solving Linear Equations

To find the inverse of a linear function like f(x) = 2x + 3, you solve for x by isolating it on one side of the equation. This involves algebraic manipulation such as subtracting constants and dividing by coefficients to express x as a function of y.

Domain and Range Considerations

When finding an inverse, it's important to consider the domain and range of the original function and its inverse. For linear functions, the domain and range are typically all real numbers, but specifying points like a = -1 helps evaluate the inverse at particular values.

In Exercises 41–44:a. Find f⁻¹(x).41. f(x) = 2x + 3, a = −1