The Inverse Function Mirror

Inverses of polynomial, radical, and rational functions

A recipe says 180 degrees Celsius. Your oven reads Fahrenheit. There is a formula to go one way. Is there always a formula to go back? For temperature, yes. For "what number did you square to get 9," the answer is 3 or −3, and that ambiguity is the whole problem.

Step 1 of 6: Two inputs, one output

Consider $f(x) = x^2$. We know $f(3) = 9$ and $f(-3) = 9$. If someone tells you the output is 9, what was the input?

Can you determine a unique input from the output 9?

Step 2 of 6: The horizontal line test

Drag the horizontal line up and down across the parabola. Count how many times it crosses the curve at each height.

Horizontal line at y = 4

For positive y-values, how many times does the line cross the curve?

Step 3 of 6: Restricting the domain

Now we restrict the domain to $x \geq 0$. Only the right half of the parabola remains. Drag the horizontal line again.

Horizontal line at y = 4

With the domain restricted, how many times does each horizontal line cross the curve?

Step 4 of 6: The reflection

Watch the restricted parabola reflect over the line $y = x$. What curve does the reflection look like?

Step 5 of 6: A cubic needs no restriction

Now consider $f(x) = x^3$. Drag the horizontal line test across this curve.

Horizontal line at y = 0

Does this function need a domain restriction to have an inverse?

Step 6 of 6: Algebra meets geometry

Consider $f(x) = \frac{2x+3}{x-1}$. To find the inverse algebraically, swap $x$ and $y$, then solve for $y$.

Start: $y = \frac{2x+3}{x-1}$

Swap: $x = \frac{2y+3}{y-1}$

Multiply: $x(y-1) = 2y+3$

Expand: $xy - x = 2y + 3$

Collect y: $xy - 2y = x + 3$

Factor: $y(x-2) = x+3$

Solve: $y = \frac{x+3}{x-2}$

Now watch the simulation reflect the original over $y = x$ and overlay the algebraic answer.

Explore Inverse Functions

Choose a function:

Show horizontal line test Show reflection over y = x Show y = x line

Points on f(x) and their swapped counterparts on the inverse
x	f(x)	Inverse x	Inverse y

Select a function and use the toggles to explore its inverse.

Choose a function above and toggle the reflection to see its inverse.

Try This

Temperature Conversion

The formula to convert Celsius to Fahrenheit is $F = \frac{9}{5}C + 32$.

Part A: What is the inverse function? (What formula converts Fahrenheit back to Celsius?)

Part B: What does the inverse function do in plain words?

Part C: Does $F^{-1}(F(100))$ give back 100?

Revenue and Units

A company's revenue is $R(x) = 50\sqrt{x - 100}$ for $x \geq 100$ units sold.

Part A: The team wants $400 in revenue. How many units do they need to sell? Use the inverse function.

Enter a whole number of units (within 1 unit).

Units needed:

Part B: Why does the domain restriction $x \geq 100$ exist?

Composing a Function With Itself

Consider $f(x) = \frac{2x+3}{x-1}$. We found in Step 6 that its inverse is $f^{-1}(x) = \frac{x+3}{x-2}$. Now compute $f(f(x))$ and see whether this function is its own inverse.

Part A: What is $f(f(x))$?

Part B: What does it mean geometrically for a function to be its own inverse?

Determine whether a function is one-to-one using the horizontal line test
Find the inverse of a function algebraically by swapping x and y
Understand that the graph of an inverse is a reflection over y = x
Restrict the domain of a non-one-to-one function to create an invertible function
Verify an inverse by checking that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Quick Check

If $f(x) = x^2$ with domain $x \geq 0$, what is $f^{-1}(25)$?

Instructor Notes

Teaching Notes

Step 1 is the conceptual anchor: two inputs producing the same output is why we need one-to-one functions before inverses exist.
Step 4 should surprise students who have not connected reflection over y = x to the algebraic swap of x and y. The point (3, 9) becoming (9, 3) is the key.
Step 6 ties the algebraic procedure to the geometric picture. Students often learn the algebra and the graphing as separate topics. This step unifies them.
The temperature conversion in Apply is intentionally mundane. It anchors the abstraction in something every student has encountered.

Common Student Errors

Confusing $f^{-1}(x)$ with $\frac{1}{f(x)}$. This is the most common error. Address it early.
Believing every function has an inverse. Step 1 and 2 are designed to break this assumption.
Forgetting to swap x and y before solving. Students often just solve for x in terms of y without swapping.
Not restricting the domain when needed. Students produce "inverses" of non-one-to-one functions.

Discussion Questions

Why does the horizontal line test determine invertibility? What is it really checking?
Can you think of a real-world process that is not reversible? What would its "function" look like?
If $f(x) = x^2$ is restricted to $x \leq 0$ instead, what is the inverse?
What other functions are their own inverse? (Hint: what does $f(x) = -x$ do?)

Exam Connection

Finding inverses algebraically (swap and solve) is a standard exam item.
Determining domains and ranges of inverse functions from the original.
Using composition to verify inverses: show $f(f^{-1}(x)) = x$.
The horizontal line test as a multiple-choice or true/false question about invertibility.