In Exercises 41–44:
a. Find f⁻¹(x).

41. f(x) = 2x + 3, a = −1

Guida verificata passo dopo passo

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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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.

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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.

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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.

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