E3.3 - Inverse functions (Extended)

Reversing a function

If $f(x)=3x-4$, the input 5 produces $f(5)=11$. Reversing this pair sends 11 back to 5: add 4, then divide by 3. Reverse the order as well as the operations.

The superscript $-1$ here names an inverse function; it does not mean a reciprocal. For the example, $f^{-1}(11)=5$, whereas $1/f(11)=1/29$.

[DIAGRAM: asset_name: Reversing three mapping arrows; asset_slug: inverse_mapping_pairs; caption: An inverse reverses every arrow and exchanges the input and output sets.; description: Left mapping shows f of x equals 2x plus 1 on the domain minus one, zero and two, with arrows to minus one, one and five respectively. Right mapping shows the inverse arrows from minus one, one and five back to minus one, zero and two. Input and output columns are explicitly labelled.]

For $f(x)=2x+1$ on the finite domain $\{-1,0,2\}$, the pairs are $-1\mapsto-1$, $0\mapsto1$, $2\mapsto5$. The inverse has domain $\{-1,1,5\}$ and sends $5\mapsto2$, for instance. A mapping diagram can therefore give inverse values without any rearrangement.

A reverse rule must still be a function: each input must have exactly one output. On the whole real line, $h(x)=x^2$ sends both 2 and $-2$ to 4, so reversing would send 4 to two different numbers. It has no inverse function on that whole domain. Restricting it to $x\ge0$ removes that ambiguity, giving the inverse $h^{-1}(x)=\sqrt{x}$ for $x\ge0$. A function whose distinct inputs always give distinct outputs is called one-to-one.

Read the inverse notation as a reversed pair.

Finding an inverse

To turn the reversed pairs into a formula, write $y=f(x)$ and rearrange to make the original input $x$ the subject. The result tells you the input in terms of the output. Rename the inverse's input $x$ only after completing that rearrangement.

Change the input operation from $x$ to $x^2$. A domain restriction is now essential when choosing the square-root sign.