Chapter 1, Problem 310RE

True or False? Justify your answer with a proof or a counterexample.

310. A function is always one-to-one.

To determine

To calculate: Justify the statement with a proof or a counterexample

“A function is always one-to-one.”

Answer to Problem 310RE

The statement “A function is always one-to-one” is false.

Explanation of Solution

Given information: Given statement is “A function is always one-to-one.”

Formula used: A function is said to be one-to-one, if every element of the range of the function corresponds to exactly one element of the domain.

For a function to be one-to one

If $f(a)=f(b)$

Then $a=b$

Calculation:

Let us consider an example.

Let $f(x)=x^2$is one to one

This implies that

$f(x_1)=f(x_2)$

$\Rightarrow {(x_1)}^2={(x_2)}^2$

$\Rightarrow x_1=\pm x_2$

Thus, for element in range, there exists two elements in domain.

Hence, the function is not always one-to-one.

Conclusion:

Hence, the statement “A function is always one-to-one” is false.