A. Characteristics of Functions · Functions and Equivalent ExpressionsInverse Functions: Undoing a Process, Tables and Reflections
A1.4 A1.5 A1.6About 70 minutes
Learning goals- I can explain an inverse as the process that undoes a function, applying inverse operations in reverse order.
- I can find the inverse of a function from a table of values or a graph.
- I can explain why the graph of an inverse is the reflection of the graph of the function in the line y = x.
- I can relate the domain and range of a function to those of its inverse and decide whether the inverse is a function.
Words to know
inverse of a function
The relation that reverses the function: if f takes a to b, the inverse takes b to a.
f⁻¹
Notation for the inverse of f when the inverse is a function. f⁻¹(x) is not 1/f(x).
inverse operations
Operations that undo each other, such as adding and subtracting, or multiplying and dividing.
reflection in y = x
The transformation (a, b) → (b, a).
horizontal line test
If a horizontal line meets the graph of f more than once, the inverse of f is not a function.
Socks, boots and secret codes
On a snowy February morning you put on your socks, then your boots. When you get home you undo this, but not in the same order: boots off first, then socks. Undoing a process means doing the opposite steps in the reverse order.
Secret codes work the same way. If a code shifts every letter 3 places forward in the alphabet (A → D, B → E ...), decoding shifts every letter 3 places back. The message CAT becomes FDW, and FDW decodes back to CAT.
A function is a process that turns an input into an output. Its inverse is the process that turns the output back into the input.
The inverse of a function
If a function f takes x to y, its inverse takes y back to x.
- If f(a) = b, then the inverse takes b to a.
- Every point (a, b) on f becomes the point (b, a) on the inverse.
When the inverse is also a function, we write it as f⁻¹ (read "f inverse"). So if f(5) = 13, then f⁻¹(13) = 5.
Warning: the −1 in f⁻¹ is not an exponent. f⁻¹(x) does not mean 1/f(x).
Worked exampleExample 1: Reversing a process
The function f takes a number, multiplies it by 4, then subtracts 7.
(a) Find f(3). (b) Describe the inverse in words. (c) Use the inverse to find the input that gives the output 33.
- (a) f(3) = 4(3) − 7 = 12 − 7 = 5.
- (b) The last step of f was "subtract 7", so the first step of the inverse is add 7. Then undo "multiply by 4": divide by 4.
- (c) Start with 33: 33 + 7 = 40, then 40 ÷ 4 = 10.
- Check by running f forwards: f(10) = 4(10) − 7 = 33. ✓
f(3) = 5; the inverse adds 7 then divides by 4; the input for output 33 is 10, so f⁻¹(33) = 10.
Numeric inverse: swap the columns
In a table of values, the inverse is found by switching the x and y columns. The inputs become outputs and the outputs become inputs. That also means:
- domain of the inverse = range of the function
- range of the inverse = domain of the function
Worked exampleExample 2: Population and its inverse
A small northern Ontario town tracks its population. t is years since 2020 and P is the population in thousands.
- t: 0, 1, 2, 3, 4
- P: 12, 13, 15, 18, 22
(a) Make a table for the inverse. (b) What question does the inverse answer? (c) Is the inverse a function?
- (a) Swap the rows. The inverse has input P and output t:
P: 12, 13, 15, 18, 22 → t: 0, 1, 2, 3, 4.
- (b) The function answers "How many people after t years?" The inverse answers "How many years until the population reaches P thousand?" For example, the inverse sends 18 to 3: the population reached 18 000 in 2023.
- (c) The populations 12, 13, 15, 18, 22 are all different, so each input of the inverse has exactly one output. Yes, the inverse is a function.
- To graph the inverse, plot (12, 0), (13, 1), (15, 2), (18, 3), (22, 4) on new axes with P on the horizontal axis.
The inverse table is P = 12, 13, 15, 18, 22 → t = 0, 1, 2, 3, 4. It is a function, with domain {12, 13, 15, 18, 22} and range {0, 1, 2, 3, 4}.
Graphical inverse: reflect in the line y = x
Swapping coordinates, (a, b) → (b, a), has a neat geometric meaning: it reflects the point in the line y = x. So the graph of the inverse is the mirror image of the graph of the function in the line y = x.
To sketch an inverse:
- Draw the dashed line y = x.
- Pick some key points on the function and swap their coordinates.
- Plot the new points and join them with the same kind of curve.
Points that lie on the line y = x (where x and y are equal) do not move.
Is the inverse a function?
The inverse of a function is always a relation, but it is not always a function. Look at y = x²: the points (2, 4) and (−2, 4) become (4, 2) and (4, −2) on the inverse. One input, two outputs.
A quick test: if a horizontal line crosses the graph of f more than once, then after reflecting, a vertical line crosses the inverse more than once. So:
- f passes the horizontal line test → the inverse is a function.
- f fails it → the inverse is not a function.
Worked exampleExample 3: A quadratic table
Find the inverse of the function given by the table and decide whether it is a function.
- x: −2, −1, 0, 1, 2
- y: 4, 1, 0, 1, 4
- Swap: the inverse has the pairs (4, −2), (1, −1), (0, 0), (1, 1), (4, 2).
- The input 4 has two outputs, −2 and 2. The input 1 has two outputs, −1 and 1.
- So the inverse is not a function.
- Domain of the inverse = range of f = {0, 1, 4}. Range of the inverse = domain of f = {−2, −1, 0, 1, 2}.
The inverse is the relation {(4, −2), (1, −1), (0, 0), (1, 1), (4, 2)}, which is not a function.
f⁻¹ only when it is a function
Use the notation f⁻¹ only when the inverse is a function. For y = x², just say "the inverse relation" or write its equation, x = y².
Just for funThe inverse of putting toothpaste back into the tube does exist mathematically. Physically, you are on your own.
Hands-on activityMirror on the line y = x
You need: grid paper, a ruler, a pencil, and tracing paper or a small mirror (a phone camera off and its dark screen works too).
- Draw axes from −3 to 6. Draw the dashed line y = x.
- Plot f(x) = 2x + 1 using (−1, −1), (0, 1), (1, 3), (2, 5).
- Swap each point's coordinates and plot the new points. Join them.
- Fold the grid paper along y = x (or trace the graph, then flip the tracing over along y = x). Do the two lines match?
- Repeat with y = x² using (−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4). Run a vertical ruler across the new graph.
What to look for: the inverse of y = 2x + 1 passes through (−1, −1), (1, 0), (3, 1), (5, 2). The point (−1, −1) is on y = x, so it stays put. The fold lines the graphs up exactly. The inverse of y = x² opens to the right, and a vertical ruler at x = 4 touches it twice.
Your turn
1A process adds 5, then multiplies by 3. Describe the inverse.
Divide by 3, then subtract 5.
2If f(2) = 9, what is f⁻¹(9)?
3A function contains the points (1, 6), (2, 8), (3, 10). List the points of its inverse.
4A function has domain {x ∈ ℝ | x ≥ 0} and range {y ∈ ℝ | y ≥ 3}. State the domain and range of its inverse.
They swap: domain {x ∈ ℝ | x ≥ 3}, range {y ∈ ℝ | y ≥ 0}.
5The point (−4, 7) is on a graph. Which point is on its reflection in y = x?
6Is the inverse of {(1, 5), (2, 5), (3, 6)} a function? Explain.
The inverse is {(5, 1), (5, 2), (6, 3)}. The input 5 has two outputs, so no.
7Does f⁻¹(x) mean 1/f(x)? Explain.
No. f⁻¹ is the inverse function, which undoes f. The −1 is not an exponent.
More practice: warm-up
1What is the inverse of "subtract 8"?
2Which line do you reflect in to get the graph of an inverse?
3Swap the coordinates of (0, −3).
4If a point is on y = x, such as (2, 2), where is its image on the inverse?
The same point, (2, 2). It does not move.
5What is the decoded word if the code shifts each letter 1 place forward and the coded word is IBU?
Shift back 1: I → H, B → A, U → T: HAT.
More practice: on your own
1f takes x, squares it, then adds 1. Can you undo this to get a unique input from the output 10? Explain.
Undo: subtract 1 to get 9, then take the square root: 3 or −3. Both work (f(3) = f(−3) = 10), so the input is not unique and the inverse is not a function unless the domain is restricted.
2A function has the table x: 0, 2, 4, 6 and y: −1, 3, 7, 11. Make the inverse table and decide whether the inverse is a function.
Inverse: x: −1, 3, 7, 11 → y: 0, 2, 4, 6. Each input has one output, so yes, it is a function.
3The graph of f is a line segment from (−2, 1) to (4, 4). Describe the graph of the inverse and state its domain and range.
Swap the endpoints: a segment from (1, −2) to (4, 4). Domain {x ∈ ℝ | 1 ≤ x ≤ 4}, range {y ∈ ℝ | −2 ≤ y ≤ 4}.
4A cell phone plan costs C(m) = 0.10m + 20 dollars for m minutes. Describe the inverse in words and use it to find how many minutes cost $45.
Inverse: subtract 20, then divide by 0.10. (45 − 20) ÷ 0.10 = 25 ÷ 0.10 = 250 minutes.
5Explain why the inverse of a horizontal line y = 3 is not a function.
Every point has y = 3, so after swapping, every point has x = 3: the inverse is the vertical line x = 3, which fails the vertical line test.
Challenge
1A temperature in °C is converted with F = 1.8C + 32. Write the inverse process in words and convert 77°F to Celsius. Then explain what the point (25, 77) and its swapped partner each mean.
Inverse: subtract 32, then divide by 1.8. (77 − 32) ÷ 1.8 = 45 ÷ 1.8 = 25°C. (25, 77) on the function means 25°C is 77°F; (77, 25) on the inverse means 77°F is 25°C.
2Find a point on f(x) = 3x − 4 that is also on its inverse. Explain how you found it.
Points that lie on y = x are on both graphs. Solve 3x − 4 = x: 2x = 4, x = 2. The point is (2, 2). Check: f(2) = 2, so (2, 2) swaps to itself.
3A function f is decreasing everywhere and its graph passes through (0, 5) and (5, 0). Explain why its inverse also passes through both points, and whether the inverse could be f itself.
Swapping (0, 5) gives (5, 0) and swapping (5, 0) gives (0, 5), so both points are on the inverse. It could be f itself: for example f(x) = 5 − x is its own inverse, because its graph is symmetric about y = x (reflecting it gives the same line).
Quiz
1. Which process is the inverse of "multiply by 5, then add 2"?
2. If (3, −8) is on the graph of f, which point is on the graph of its inverse?
3. The graph of an inverse is a reflection of the original graph in which line?
4. A function has domain {x ∈ ℝ | x ≥ 2} and range {y ∈ ℝ | y ≤ 5}. What is the domain of its inverse?
5. For which function is the inverse not a function?
6. If f(4) = 10, what is f⁻¹(10)?
Big idea: Expectations A1.4, A1.5 and A1.6 connect inverses to "undoing" a process, ask students to find an inverse from a table or graph, to see that the inverse's graph is the reflection in y = x, and to relate the domain and range of a function to those of its inverse.
How to help: Practise undoing everyday processes in the right order (wrapping a gift, a recipe, a code). Then ask: "If f(2) = 9, what does the inverse do to 9?"
Common difficulties:
- Undoing steps in the same order instead of the reverse order.
- Reading f⁻¹(x) as 1/f(x).
- Reflecting in the x-axis or y-axis instead of y = x.
- Assuming the inverse is always a function (it fails whenever two inputs of f share an output, as with y = x²).
- Forgetting that domain and range swap.