Question

For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function. $f(x)=x+5$

Name: Problem 49, Chapter 9: Intermediate Algebra for College Students
Uploaded: 2021-09-08T14:02:48-08:00
Duration: 1 min 4 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+5

   For each function, a) determine whether it is one-to-one; b) if it is one-to-one, determine its inverse function.
$f(x)=x+5$

Intermediate Algebra for College Students

Allen R. Angel;… 10th Edition

Chapter 9, Problem 49

Instant Answer

Solved by Expert Yujie Wang

09/08/2021

Step 1

In other words, if $f(x_1) = f(x_2)$, then $x_1$ must equal $x_2$. Let's check this for the given function: $f(x_1) = x_1 + 5$ $f(x_2) = x_2 + 5$ If $f(x_1) = f(x_2)$, then: $x_1 + 5 = x_2 + 5$ Subtracting 5 from both sides, we get: $x_1 = x_2$ Since $x_1$ 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. $f(x)=x+5$

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Key Concepts

One-to-One Functions

A one-to-one function, or an injective function, is defined by the property that no two different inputs produce the same output. This concept is fundamental in understanding how functions behave, ensuring that each element of the function's domain corresponds to a unique element in its range, which is essential for the function to be reversible.

Inverse Functions

The inverse function reverses the process of the original function so that if f(x) produces a certain value, the inverse function, denoted as f?¹(x), returns the original input from that value. To determine the inverse, one typically swaps the dependent and independent variables and solves for the new dependent variable. This process is only valid when the function is one-to-one, ensuring that every output uniquely corresponds to an input.

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