How to tell whether a function has an inverse that is also a function

Key concept: When a function has an inverse that is also a function, the key idea is that the original function must not repeat outputs for different inputs. That is the real test students should keep in mind.

Step by step

Let's go step by step.

For a function to have an inverse that is also a function, the original function must be one-to-one.

Why?

• If two different inputs give the same output, then the inverse would have one input paired with two outputs.
• That would fail the definition of a function.

For $f(x)=2x-3$:

• It is a linear function with nonzero slope.
• So every output comes from exactly one input.
• Therefore, it is one-to-one, and its inverse relation is also a function.

Correct choice

$f(x)$ is a one-to-one function.

You can also check algebraically:

$y=2x-3$

Swap $x$ and $y$:

$x=2y-3$

Solve for $y$:

$2y=x+3 \quad\Rightarrow\quad y=\frac{x+3}{2}$

That inverse is a function because each input produces exactly one output.

Pitfall alert

A common mistake is choosing the vertical line test. That test only tells you whether the original graph is a function, not whether its inverse is a function. For inverse functions, the useful idea is the horizontal line test or, more directly, one-to-one behavior.

Try different conditions

If the function were something like $f(x)=x^2$, the inverse relation would not be a function over all real numbers because both $2$ and $-2$ map to $4$. But if you restrict the domain to $x\ge 0$, then the inverse becomes a function.

Further reading

one-to-one function, inverse relation, horizontal line test