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
Last stop you chained two rules: the output of one fed the next, and you saw that fg(x) and gf(x) usually differ. An inverse is a very special second rule. It is the rule that undoes the first. Feed any input through f and then through f to the power of -1, of x, and you are back where you started. That minus one is a label, not a power: say the symbol as 'f inverse'. The distinction carries its weight in this topic.
How do you find the inverse of a linear rule? Swap and rearrange. Start from y = 2x + 1. Swap the x and the y to get x = 2y + 1. Now rearrange to make y the subject, exactly as you did with formulae: 2y = x - 1, so y = (x - 1) / 2. That new rule is the inverse. The swap is the whole idea in one move. In the original, x is the input and y is the output. After the swap, the old output is the input, which is exactly what an undoing should do.
The check is a composition. Feed f into its inverse and you should get x back, and the other way round too. If f of f inverse of x, gives x, the rearranging was right. If it gives something else, hunt for the slip. The classic trap: f inverse of x, is not 1 over f of x. The minus one is a label, not an exponent, so the inverse is not the reciprocal. The reciprocal of 2x + 1 is a fraction, 1 over (2x + 1). The inverse is (x - 1) / 2. Different animals, one notation.
Find the inverse of f(x) = 2x + 1, and check it. Write y = 2x + 1. Swap: x = 2y + 1. Rearrange: 2y = x - 1, so y = (x - 1) / 2, the inverse rule. Check both ways: f of the inverse of x is 2 times (x - 1) / 2 plus 1, which is x. And the inverse of f of x is (2x + 1 - 1) / 2 = x. Both come back to x, so the check passes. A quick pipeline: f(4) is 9, and the inverse of 9 is (9 - 1) / 2 = 4.
Now a rule with a fraction. Find the inverse of f(x) = (x - 4) / 3. Write y = (x - 4) / 3. Swap: x = (y - 4) / 3. Rearrange: multiply by 3 to get 3x = y - 4, then add 4: y = 3x + 4. Check: f of the inverse of x is (3x + 4 - 4) / 3, which is 3x / 3 = x. Notice the pattern. The original subtracts 4, then divides by 3. The inverse multiplies by 3, then adds 4. The operations come back in reverse order, each undone in its turn.
Finally, the picture. The graph of the inverse is the reflection of the original in the line y = x. Why: swapping x and y trades each point's coordinates, and that is exactly what a mirror in y = x does to a point. The point (0, 1) on the graph of f becomes (1, 0) on the graph of the inverse, and (2, 5) becomes (5, 2). A straight line reflects to a straight line, so the inverse of a linear rule is always linear. Curves reflect too, but their shapes change, so the swap method is the one that keeps the work exact.
Find the inverse of a linear rule by swapping x and y and rearranging to make y the subject, check it by composing it with the original to get x back, and remember that the graph reflects in the line y = x.
2 · Watch
3 · Play
Move a point along the line and watch its twin move on the mirror image.
Take it off screen
Where it sits
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Where this leads
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24 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.