Are one-to-one functions either always increasing or always decreasing

Key concept: A one-to-one function means different inputs give different outputs. That condition is about uniqueness, not necessarily about monotonic behavior.

Step by step

No. A one-to-one function does not have to be always increasing or always decreasing.

Why not?

A function is one-to-one if it never gives the same output for two different inputs. That only means it passes the horizontal line test.

But a function can be one-to-one even if it changes direction, as long as it never repeats an output value.

Example

Consider a function defined on a restricted domain, such as a curve that goes up, then down, but still never takes the same $y$-value twice on that domain. It is one-to-one, but not monotonic.

Important distinction

• Increasing/decreasing describes how the function changes as $x$ increases.
• One-to-one describes whether output values are repeated.

So the correct answer is:

$\boxed{\text{No, one-to-one functions are not always increasing or always decreasing.}}$

Pitfall alert

Do not confuse injective behavior with monotonic behavior. A function can be one-to-one on a limited domain without being globally increasing or decreasing.

Try different conditions

For many functions on an interval, strict monotonicity does imply one-to-one. For example, $f(x)=x^3$ is strictly increasing and one-to-one. But the converse is not always true unless extra conditions are given.

Further reading

one-to-one, horizontal line test, monotonic