Inverse Functions: Undo the Steps in Reverse Order

The question

The function $f$ is defined by

f(x) = \frac{3x - 4}{2}

for all real values of $x$ .

(a) Find an expression for $f^{-1}(x)$ . [2 marks]

(b) Find the value of $f^{-1}(7)$ . [1 mark]

The common wrong answer

Many students try to find the inverse by “undoing” each step of $f$ in their head. They read $f$ as: multiply by 3, subtract 4, divide by 2. Then they undo each step in the same order: divide by 3, add 4, multiply by 2.

f^{-1}(x) = 2\left(\frac{x}{3} + 4\right) = \frac{2x}{3} + 8

Everything after that is built on this wrong inverse:

f^{-1}(7) = \frac{2(7)}{3} + 8 = \frac{38}{3}

and for part (c):

\begin{aligned} \frac{3x - 4}{2} &= \frac{2x}{3} + 8 \\ 9x - 12 &= 4x + 48 \\ x &= 12 \end{aligned}

The mistake is the order. Undoing a function is like taking off your shoes and socks: the last thing you put on is the first thing you take off. The last step of $f$ is “divide by 2”, so the first step of $f^{-1}$ must be “multiply by 2”. The Cambridge 0606 examiner report for June 2024 noted this in many responses: an incorrect order of operations was used to find the inverse function.

A one-line check catches it. If $f^{-1}(7)$ really were $\frac{38}{3}$ , then $f\left(\frac{38}{3}\right)$ would have to be 7. But

f\left(\frac{38}{3}\right) = \frac{3 \times \frac{38}{3} - 4}{2} = \frac{38 - 4}{2} = 17

which is not 7, so the “inverse” is wrong.

The correct working

(a) Write $y = f(x)$ and rearrange to make $x$ the subject. Each line undoes the last operation still left on the right-hand side.

\begin{aligned} y &= \frac{3x - 4}{2} \\ 2y &= 3x - 4 \\ 2y + 4 &= 3x \\ x &= \frac{2y + 4}{3} \end{aligned}

So $f^{-1}(y) = \frac{2y + 4}{3}$ . Write it with $x$ as the variable:

f^{-1}(x) = \frac{2x + 4}{3}

The answer is $f^{-1}(x) = \dfrac{2x + 4}{3}$ .

Check: $f^{-1}f(x) = \dfrac{2 \times \frac{3x - 4}{2} + 4}{3} = \dfrac{3x - 4 + 4}{3} = x$ . It works.

(b) Substitute 7:

f^{-1}(7) = \frac{2(7) + 4}{3} = \frac{18}{3} = 6

So $f^{-1}(7) = 6$ . Check: $f(6) = \frac{3(6) - 4}{2} = \frac{14}{2} = 7$ . Correct.

(c) Set the two expressions equal and multiply both sides by 6 to clear the fractions:

\begin{aligned} \frac{3x - 4}{2} &= \frac{2x + 4}{3} \\ 3(3x - 4) &= 2(2x + 4) \\ 9x - 12 &= 4x + 8 \\ 5x &= 20 \\ x &= 4 \end{aligned}

So $x = 4$ . Check: $f(4) = \frac{12 - 4}{2} = 4$ and $f^{-1}(4) = \frac{8 + 4}{3} = 4$ . Both give 4.

Graphs of y = f(x) and y = f inverse of x with the dashed line y = x. The two lines are reflections of each other in y = x and meet at (4, 4). The point (6, 7) lies on f and the point (7, 6) lies on f inverse. — The graph of the inverse is the reflection of f in the line y = x, so the two graphs meet on that line at (4, 4).

Why it works

A function is a machine that does steps in order. Here $f$ takes $x$ , multiplies by 3, subtracts 4, then divides by 2. To get back to $x$ , you have to walk the same path backwards: multiply by 2, add 4, then divide by 3. That is exactly $\frac{2x + 4}{3}$ .

Rearranging $y = f(x)$ does this for you automatically, because you always peel off the outermost operation first. That is why it is safer than doing it in your head.

The graph shows the second idea used in part (c). The inverse swaps every input and output, so $(6, 7)$ on $f$ becomes $(7, 6)$ on $f^{-1}$ . Swapping coordinates is a reflection in $y = x$ . For an increasing function like this one, $f$ and $f^{-1}$ can only meet on the line $y = x$ , so you could also have solved part (c) with $f(x) = x$ , which gives $3x - 4 = 2x$ , so $x = 4$ .

The wrong inverse fails this test too: it meets $f$ at $(12, 16)$ , which is not on $y = x$ .

How to spot it next time

Find an inverse by writing $y = f(x)$ and rearranging line by line, not by reversing steps in your head.
Check your answer with one number: pick a value such as $x = 6$ , work out $f(6) = 7$ , and make sure your $f^{-1}(7)$ gives 6 back.
In “solve $f(x) = f^{-1}(x)$ ” questions, your answer should satisfy $f(x) = x$ as well. If it doesn’t, recheck your inverse.