Question

Verbal

4. Are one-to-one functions either always increasing or
always decreasing? Why or why not?

Answer 1

If a function is continuous and one - to - one then it is either always increasing or always decreasing.

Explanation:

An easy way to see this on a graph is to draw a horizontal line through the graph . If the line only cuts the curve once then the function is one - to - one.

Answer 2

If a one-to-one function is not continuous, it is sometimes increasing or decreasing. If a one-one function is continuous, it is always increasing or always decreasing.

For example, if
`$f(x)=\left\{\begin{array}{lll}-x & \text { if } & |x| \geq 1 \\ x & \text { if } & |x| < 1\end{array}\right.$`

This function is one-to-one and it is not always increasing or decreasing.

Explanation:

A statement that are one-to-one functions either always increasing or always decreasing is given.

It is required to explain the given condition.

To explain the given condition, consider a continuous and a not continuous function then explain about the one-to-one function.

If a one-to-one function is not continuous, it is sometimes increasing or decreasing.

For example, if
`$f(x)=\left\{\begin{array}{lll}-x & \text { if } & |x| \geq 1 \\ x & \text { if } & |x| < 1\end{array}\right.$`

This function is one-to-one and it is not always increasing or decreasing.

If a one-one function is continuous, it is always increasing or always decreasing.

For example, if f(x)=x-3. It is a continuous function and it reaches a minimum value at some values of x and it then keeps increasing.

There will be the same output value for different input values. It does not pass the horizontal line test and so it is not a one-to-one function.