Question

In Problem 8, (a) verify that the function is one-to-one, and (b) find the inverse of the given function.

$\left\{\left(1,2\right),\left(3,5\right),\left(5,8\right),\left(6,10\right)\right\}$

Short answer

(a) The function is one-to-one .

(b) The inverse of the function is

$\left\{\left(2,1\right),\left(5,3\right),\left(8,5\right),\left(10,6\right)\right\}$.

Step-by-step solution

Part(A) Step 1 : Definition

A function is one-to-one if any two different inputs in the domain correspond to two different outputs in the range.

Part(a) Step 2: Concept used

By definition, if $x_1$and $x_2$are two different inputs of a function $f$, then function $f$is one-to-one if $f\left(x_1\right)\ne f\left(x_2\right)$.

Given that,

$\left\{\left(1,2\right),\left(3,5\right),\left(5,8\right),\left(6,10\right)\right\}$

There are no two distinct inputs that correspond to the same output.

So, the function is one-to-one .

Part(b) Step 1: Definition

Suppose that $f$is a one-to-one function. Then, to each $x$in the domain of $f$, there is exactly one $y$in the range, and to each $y$in the range of $f$, there is exactly one $x$in the domain.

The correspondence from the range of $f$back to the domain of $f$is called the inverse function of $f$.

Part(b) Step 2: Concept used

Given that

$\left\{\left(1,2\right),\left(3,5\right),\left(5,8\right),\left(6,10\right)\right\}$

The inverse of the given function is found by interchanging the entries in each ordered pair.

So, the inverse of the given function be

$\left\{\left(2,1\right),\left(5,3\right),\left(8,5\right),\left(10,6\right)\right\}$

The domain of the function is role="math" localid="1646324991787" $\left\{1,3,5,6\right\}$.

The range of the function is role="math" localid="1646324955812" $\left\{2,5,8,10\right\}$.

The domain of inverse function is $\left\{2,5,8,10\right\}$.

The range of inverse function is $\left\{1,3,5,6\right\}$.