The functions in Exercises 11-28 are all one-to-one. For each function, a. Find an equation for f^-1(x), the inverse function. b. Verify that your equation is correct by showing that f(ƒ^-1 (x)) = = x and ƒ^-1 (f(x)) = x. f(x) = 1/x

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

Next, interchange the variables \(x\) and \(y\) to find the inverse: \(x = \frac{1}{y}\).

Now, solve this equation for \(y\) to express the inverse function explicitly. Multiply both sides by \(y\) to get \(xy = 1\), then divide both sides by \(x\) to isolate \(y\): \(y = \frac{1}{x}\).

Thus, the inverse function is \(f^{-1}(x) = \frac{1}{x}\). Notice that the function is its own inverse.

To verify, check the compositions: compute \(f(f^{-1}(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x\) and \(f^{-1}(f(x)) = f^{-1}\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x\). Both compositions return \(x\), confirming the inverse is correct.

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Key Concepts

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

One-to-One Functions

A one-to-one function assigns each input exactly one unique output and vice versa, ensuring it has an inverse. This property is essential because only one-to-one functions have inverses that are also functions, allowing us to reverse the mapping from outputs back to inputs.

Inverse Functions

The inverse of a function f, denoted f⁻¹, reverses the effect of f, swapping inputs and outputs. To find f⁻¹(x), solve the equation y = f(x) for x in terms of y, then interchange variables. The inverse satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.

Verification of Inverse Functions

To confirm that two functions are inverses, compose them in both orders: f(f⁻¹(x)) and f⁻¹(f(x)). Both compositions must simplify to x for all x in the domains. This step ensures the correctness of the inverse function found.

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