Question

For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function. $t(x)=-|x|$

Name: Problem 54, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2021-09-08T14:13:19-08:00
Duration: 1 min 18 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. t(x)=-|x|

   For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.
$t(x)=-|x|$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 54

Instant Answer

Solved by Expert Yujie Wang

09/08/2021

Step 1

In other words, if $t(x_1) = t(x_2)$, then $x_1$ must equal $x_2$. For the function $t(x) = -|x|$, let's consider two different input values, $x_1 = 2$ and $x_2 = -2$. We have: $t(x_1) = -|2| = -2$ $t(x_2) = -|-2| = -2$ Since $t(x_1) = t(x_2)$ but $x_1 \neq 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. $t(x)=-|x|$

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Key Concepts

One-to-One Functions

A function is one-to-one (injective) if every element of the function's domain maps to a unique element in its codomain. This means that no two distinct inputs produce the same output, which can often be verified using the Horizontal Line Test on its graph.

Inverse Functions

An inverse function essentially reverses the mapping of the original function, swapping the roles of inputs and outputs. It exists only if the original function is one-to-one, allowing each output to refer back to a unique input, which is critical for the function’s invertibility.

Absolute Value Function

The absolute value function, defined as |x|, returns the non-negative value of a number. It is characterized by its symmetry about the y-axis, which can affect properties like injectivity because both positive and negative inputs yield the same output.

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