# The condition for the slope of a linear function to be one-to-one. To find the inverse of a linear one-to-one function and its slope. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.7, Problem 90E
To determine

## To find: The condition for the slope of a linear function to be one-to-one. To find the inverse of a linear one-to-one function and its slope.

Expert Solution

A linear function f(x)=mx+b be one-to-one if its slope is not equal to 0 i.e. (m0) .

The inverse of the given f(x) is f(x)=1mx+(bm)

Yes, the inverse of the given function linear.

The slope of the inverse function is 1m .

### Explanation of Solution

Given information: A linear function f(x)=mx+b

Concept used:

The linear function can be written as f(x)=mx+b where m= Slope .

A function with domain A is called a one-to one function if no two elements of A have the same image, that is f(x1)f(x2)     whenever   x1x2

To be one-to-one the function should not have any repeated value horizontally.

If f be a one-to-one function with domain A and range B. Then its inverse function f1 has domain B and range A and is defined by

f1(y)=x      f(x)=y

Calculation:

The linear function is f(x)=mx+b , where m= Slope .

To be one-to-one the function should not have any repeated value horizontally.

Since, the given function is linear; to be one-to-one it should not be a line parallel to X-axis.

The slope of a line parallel to x-axis is 0 .

So, to be a one-to-one function the slope have to be not equal to 0

(m0) .

Let, f(x)=mx+b is one-to-one i.e. (m0) .

f(x)=mx+b          (m0)y=mx+bmx=ybx=ybmf(y)=ybmf(x)=xmbmf(x)=1mx+(bm)

Therefore, the inverse of the given f(x) is

f(x)=1mx+(bm)

Yes, the inverse of the given function linear as it can be written in the form of f(x) .

The slope of the inverse function is 1m .

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