The Inverse of a Function

At the boards the puzzle ran backwards: “my function multiplied a number by 3, then added 2, and printed 17 — what went in?” Every group undid it the same way — subtract 2, then divide by 3 — inverse operations, in reverse order, exactly how you take off shoes and socks. The inverse of a function is that undoing, packaged as a function of its own and written $f^{-1}$.

Undoing, in the right order

For $f(x)=3x+2$, the inverse should take outputs back to inputs. The algebra mirrors the idea:

• Write the function as $y=3x+2$.
• Swap $x$ and $y$ — outputs become inputs: $x=3y+2$.
• Solve for $y$, applying inverse operations in reverse order: $y=\frac{x-2}{3}$.
• Test one pair: $f(5)=17$, so $f^{-1}(17)$ had better be 5. It is.

That last box is not optional politeness — it is Checking Your Own Work doing its cheapest, fastest job.

The mirror line

Swapping inputs and outputs has a picture. Take any table of values for $f$, exchange its columns, and you have a table for the inverse; plot both and every point $(a,b)$ faces a partner $(b,a)$ across the line $y=x$. The graph of the inverse is the reflection of the graph of $f$ in that line — tracing paper folded along $y=x$ shows it in one move, and Using Desmos confirms it in two. The swap also trades the sets from Domain and Range: the domain of $f$ becomes the range of $f^{-1}$, and vice versa.

When the inverse is not a function

Reflect $f(x)=x^2$ in the mirror line and the image fails the vertical-line test — the output 9 came from both 3 and $-3$, so the undoing cannot decide where to send 9 back. The inverse relation always exists; it is a function only when $f$ never repeats an output. You can rescue $x^2$ by restricting its domain: keep $x\ge 0$ and the inverse is $g(x)=\sqrt{x}$; keep $x\le 0$ and it is $h(x)=-\sqrt{x}$.

The same story in algebra, for $f(x)=(x-2{)}^2-5$: swapping and solving gives $y=2\pm \sqrt{x+5}$ — the $\pm$ is the algebra confessing that two answers exist.

Inverses are one of the four wings of The Transformation Gallery, so your curated examples for that task are the natural place to practise these moves on functions you chose yourself.

Curriculum connection

A1.4

relate the process of determining the inverse of a function to their understanding of reverse processes (e.g., applying inverse operations)

Link to original

A1.5

determine the numeric or graphical representation of the inverse of a linear or quadratic function, given the numeric, graphical, or algebraic representation of the function, and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the graph of a function and the graph of its inverse (e.g., the graph of the inverse is the reflection of the graph of the function in the line $y=x$) Sample problem: Given a graph and a table of values representing population over time, produce a table of values for the inverse and graph the inverse on a new set of axes.

Link to original

A1.6

determine, through investigation, the relationship between the domain and range of a function and the domain and range of the inverse relation, and determine whether or not the inverse relation is a function Sample problem: Given the graph of $f(x)=x^2$, graph the inverse relation. Compare the domain and range of the function with the domain and range of the inverse relation, and investigate connections to the domain and range of the functions $g(x)=\sqrt{x}$ and $h(x)=-\sqrt{x}$.

Link to original

A1.7

determine, using function notation when appropriate, the algebraic representation of the inverse of a linear or quadratic function, given the algebraic representation of the function [e.g., $f(x)=(x-2{)}^2-5$], and make connections, through investigation using a variety of tools (e.g., graphing technology, Mira, tracing paper), between the algebraic representations of a function and its inverse (e.g., the inverse of a linear function involves applying the inverse operations in the reverse order) Sample problem: Given the equations of several linear functions, graph the functions and their inverses, determine the equations of the inverses, and look for patterns that connect the equation of each linear function with the equation of the inverse.

Link to original