Inverse functions
Find the rule that undoes a function by swapping input and output and rearranging, and see the inverse graph as a reflection in the line y = x.
What a learner can do afterwards
- Find f⁻¹(x) for a linear rule by rearranging
- Check an inverse by showing that ff⁻¹(x) gives back x
- Describe the inverse graph as a reflection of the original in y = x
1 · Read
You have composed functions by feeding one output into another rule. An inverse undoes a function: applying f and then f⁻¹ returns the original input. Read f⁻¹ as 'f inverse'. The -1 in this notation names the inverse; it does not mean take a reciprocal. An inverse function exists only when each output of f comes from one input.
For a linear rule with nonzero slope, find the inverse by swapping input and output, then rearranging. Write y = 2x + 1. Swap x and y: x = 2y + 1. Subtract 1 to get 2y = x - 1, then divide by 2: y = (x - 1) / 2. Thus f⁻¹(x) = (x - 1) / 2. The old output becomes the new input.
Check a candidate inverse by composing both ways. You need f(f⁻¹(x)) = x for outputs of f and f⁻¹(f(x)) = x for inputs of f. A mismatch reveals an error. The inverse is different from the reciprocal: for f(x) = 2x + 1, f⁻¹(x) = (x - 1) / 2, whereas 1/f(x) = 1/(2x + 1). A check at one number tests that number, not every input.
For f(x) = 2x + 1, swap y = 2x + 1 to x = 2y + 1. Rearranging gives f⁻¹(x) = (x - 1) / 2. Check both ways: f(f⁻¹(x)) = 2((x - 1)/2) + 1 = x, and f⁻¹(f(x)) = (2x + 1 - 1)/2 = x. At input 4, f gives 9 and the inverse gives (9 - 1)/2 = 4.
For f(x) = (x - 4)/3, write y = (x - 4)/3 and swap: x = (y - 4)/3. Multiply by 3, then add 4: 3x = y - 4, so f⁻¹(x) = 3x + 4. Check f(f⁻¹(x)) = (3x + 4 - 4)/3 = x. The original subtracts 4 and divides by 3. The inverse undoes these operations in reverse order: multiply by 3, then add 4.
Axes cross at the origin (0, 0): x horizontal, y vertical. Plot (a, b) from there: a right, then b up; negatives mean left or down. Use equal unit lengths on both axes. The line y = x joins equal-coordinate points. Folding along it swaps (a, b) to (b, a). On f, a is input and b output; swapping gives the inverse graph. For f(x) = 2x + 1, (0, 1) becomes (1, 0), and (2, 5) becomes (5, 2). Nonconstant linear functions have linear inverses. Curves reflect this way when an inverse function exists.
The inverse of f⁻¹ is f. To recover f from its inverse, apply the same swap-and-rearrange method to the supplied inverse rule. Applying one inverse twice is different: evaluate the inverse once, then use that result as its next input. For h(x) = (x + 6)/7, h(8) = 2 and h(h(8)) = h(2) = 8/7.
Find an inverse by swapping x and y and solving for y. Check composition both ways on the relevant inputs. The inverse graph reflects in y = x, swapping each point's coordinates.
The practice questions and readings were written for this dataset. The referenced videos are by Khan Academy.
More to read
Which rules can be undone?
Suppose two different inputs produce the same output. An inverse receiving that output could not choose one original input, so it would not be a function.
A function is one-to-one when each output comes from just one input. On its graph, a horizontal line then meets at most one point. This is the horizontal line test.
The inverse's allowed inputs are the original function's outputs. Each assignment reverses: if f(a) = b, then f⁻¹(b) = a. Its graph therefore reflects in y = x.
Recovering the input
For p(x) = 7x - 4, an output y satisfies y = 7x - 4. Adding 4 and dividing by 7 recovers the input: x = (y + 4)/7. Naming the new input x gives p⁻¹(x) = (x + 4)/7.
The swap-and-rearrange method expresses the same reversal. Start with y = 7x - 4, swap to x = 7y - 4, then solve for y.
The inverse is not 1/(7x - 4). That fraction is a reciprocal; it does not undo the original operations.
Proving the two-way check
For p(x) = 5x + 4 and q(x) = (x - 4)/5 on the real numbers, calculate both compositions.
p(q(x)) = 5((x - 4)/5) + 4 = x. q(p(x)) = (5x + 4 - 4)/5 = x.
Both identities hold for every real input, so these functions are mutual inverses. For restricted domains, each composition must be defined and return its input on the relevant domain.
A numerical round trip is a useful check of one value. It does not replace the general identities.
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3 · Play
Move a point along the line and watch its twin move on the mirror image.
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35 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.