Chapter 5.2, Problem 13AYU

In Problems 13-20, determine whether the function is one-to-one.

To determine

To find: Determine whether the function is one-to-one.

Answer to Problem 13AYU

Solution:

No, the given function is not one – to – one function.

Explanation of Solution

Given:

Calculation:

A function is one-to-one if any two different inputs in the domain correspond to two different outputs in the range. That is, if $x_1\text{and}{x}_2$ are two different inputs of a function $f$, then $f$ is one-to-one if $f(x_1)\ne f(x_2)$.

In other way, a function $f$ is one-to-one if no $y$ in the range is the image of more than one $x$ in the domain. A function is not one-to-one if any two (or more) different elements in the domain correspond to the same element in the range.

In the domain, we have 4 different inputs and also in the range, 3 different outputs, if we map the function, we should get even different values as given in the data.

But two inputs of the domain have the same output of the range. That is 20 Hours and 25 Hours have $\$200$ as its range. As the consequence, the given function is not one-to-one, because two different elements in the domain correspond to the same element in the range.

Therefore $f$ is not one-to-one and as if $f(x_1)=f(x_2)=\$200$.