What is the inverse of x squared?
Understand the Problem
The question is asking for the inverse of the function f(x) = x^2. The approach to solve this involves finding a function that undoes the squaring operation.
Answer
The inverse of the function is \(f^{-1}(x) = \sqrt{x}\) for \(x \geq 0\).
Answer for screen readers
The inverse of the function (f(x) = x^2) is (f^{-1}(x) = \sqrt{x}) for (x \geq 0).
Steps to Solve
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Identify the Function The given function is (f(x) = x^2). This function maps (x) to (x^2).
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Set Up the Equation for Inverse To find the inverse, we need to set (y = f(x)). Thus, we have: $$y = x^2$$
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Solve for (x) in Terms of (y) Rearranging the equation (y = x^2) to solve for (x): $$x = \sqrt{y}$$ However, since we are working with the inverse function, we consider both the positive and negative roots: $$x = \pm\sqrt{y}$$
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Express the Inverse Function Thus, the inverse function can be expressed as: $$f^{-1}(y) = \sqrt{y} \text{ (for } y \geq 0)$$ Keep in mind that the original function is not one-to-one without restricting the domain, so we typically restrict (x \geq 0).
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Replace (y) Back with (x) To create the inverse function in standard notation: $$f^{-1}(x) = \sqrt{x} \text{ (for } x \geq 0)$$
The inverse of the function (f(x) = x^2) is (f^{-1}(x) = \sqrt{x}) for (x \geq 0).
More Information
The inverse function essentially reverses the operation of squaring a number. Since squaring is not a one-to-one function without restrictions, we define the inverse only for non-negative values.
Tips
- Forgetting to restrict the domain when computing the inverse of (f(x) = x^2). Remember that the inverse should only apply for (x \geq 0) to ensure it is a function.
