Question

For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function. $f(x)=x^{2}-3, x \geq 0$

Name: Problem 64, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2021-09-09T04:32:37-08:00
Duration: 38 s
Channel: Yujie Wang
Description: For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function. f(x)=x^2-3, x ≥0

   For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.
$f(x)=x^{2}-3, x \geq 0$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 64

Instant Answer

Solved by Expert Yujie Wang

09/09/2021

Step 1

Since the function is defined for $x \geq 0$, we can see that it is one-to-one. This is because for non-negative values of x, the function $x^2$ is always increasing, and therefore, each input value will have a unique output value. b) Show more…

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For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function. $f(x)=x^{2}-3, x \geq 0$

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Key Concepts

Inverse Function

An inverse function reverses the effect of the original function, meaning that the inverse function will map the output back to the original input. For a function to have an inverse, it must be one-to-one so that every output corresponds to exactly one input.

Domain Restriction

Domain restrictions are often applied to functions to ensure that they are one-to-one. By limiting the function's domain appropriately, especially in cases of functions like quadratic functions which are naturally not one-to-one, we can guarantee that each output will come from exactly one input, making it possible to define an inverse.

One-to-One Function

A one-to-one function, or injective function, is one in which every element of the function’s domain maps to a unique element in its range. This means that no two distinct inputs produce the same output, which is a necessary condition for the existence of an inverse function.

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