# The definition of one-to-one function. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1, Problem 12RCC

(a)

To determine

## To explain: The definition of one-to-one function.

Expert Solution

### Explanation of Solution

A function f is called one-to-one if every element of the range corresponds to exactly one element of the domain.

That is, if f(x1)f(x2), then x1x2.

In other words, a function is said to be one to one if every image as a unique pre image in the domain.

Graph is one-to-one if and only if every horizontal line intersect function f at most once.

Therefore, horizontal line must cross at only one point or it should be parallel to the graph.

(b)

To determine

### To define: The inverse function f−1 for a one to one function f and explain the construction of the graph of the inverse f−1.

Expert Solution

Solution:

If the function f is one-to-one then the graph of f and graph of f1 are reflections of each other about the line y=x.

### Explanation of Solution

Given:

Function f is a one-to-one function.

Let y=f(x) be the given function.

Replace x by y x=f(y).

Take f1 on both sides.

f1(x)=f1(f(y))f1(x)=y

Check whether the graph of the function f is one-to-one by the horizontal line test in order to find the graph of f1.

If the function f is not one-to-one then f1 does not exist.

If the function f is one-to-one then the graph of f and graph of f1 are reflections of each other about the line y=x.

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