Why Only a One-One and Onto Function Can Be Reversed

A function turns inputs into outputs, but not every function can be run backwards. If two inputs share an output, or some outputs are never reached, there is no single way back. Classifying functions shows exactly when an inverse exists.

This lesson covers one-one and onto functions, invertibility, composite and inverse functions, and graphs of a function with its inverse.

**A function $f:A\to B$ is one-one if different inputs always give different outputs and many-one otherwise, and it is onto if every element of B is the image of some input and into if some element of B is never reached.

Tests:

- One-one** — show $f(x_1)=f(x_2)\Rightarrow x_1=x_2$; on a graph, every horizontal line meets the curve at most once
- Onto — for every y in B, solve $f(x)=y$ and check that x lies in A
- Bijective — both one-one and onto

Worked example. $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=3x-5$.

- One-one: $3x_1-5=3x_2-5$ gives $x_1=x_2$
- Onto: for any real y, $x=\frac{y+5}{3}$ is real and $f(x)=y$
- So f is bijective

Worked example 2. $g:\mathbb{R}\to \mathbb{R}$ with $g(x)=x^2$ is many-one, since $g(-2)=g(2)=4$, and into, since no real x gives $g(x)=-1$.

An everyday example. Giving each student in a class of 40 a different roll number from 1 to 40 is a bijection — no two students share a number and every number is used.

The substance. The same rule can change type when the sets change — $x^2$ from $[0,\infty )$ to $[0,\infty )$ is both one-one and onto.

**A function $f:A\to B$ is invertible exactly when it is bijective — one-one, so that each output comes from a single input, and onto, so that every element of B has an input to return to.

Why both conditions are needed:**

- If f is not one-one, the output 4 of $x^2$ would have to return to both 2 and $-2$
- If f is not onto, an unreached element of B would have nowhere to go

The inverse. For a bijective f, $f^{-1}:B\to A$ is defined by $f^{-1}(y)=x$ exactly when $f(x)=y$, so $f^{-1}(f(x))=x$ and $f(f^{-1}(y))=y$.

Worked example. Is $f:\mathbb{N}\to \mathbb{N}$ with $f(n)=2n$ invertible? It is one-one, since $2n_1=2n_2$ gives $n_1=n_2$, but it is not onto, since 3 is not an image. So it is not invertible as a function into $\mathbb{N}$.

Restricting the domain. $x^2$ becomes invertible from $[0,\infty )$ to $[0,\infty )$, with inverse $\sqrt{x}$.

An everyday example. A locker system that gives each student exactly one locker and leaves no locker empty can be reversed — from any locker you can name its owner.

The substance. Invertibility depends on the chosen codomain — $f(n)=2n$ is invertible as a function from $\mathbb{N}$ onto the even natural numbers.

**The composite $g\circ f$ is defined by $(g\circ f)(x)=g(f(x))$ — apply f first, then g — and the inverse of an invertible function is found by writing $y=f(x)$, solving for x in terms of y, and renaming the variable.

Worked example (composite).** Let $f(x)=2x+1$ and $g(x)=x^2$.

- $(g\circ f)(x)=(2x+1{)}^2$, so $(g\circ f)(3)=49$
- $(f\circ g)(x)=2x^2+1$, so $(f\circ g)(3)=19$
- Since $49\ne 19$, composition is not commutative

Worked example (inverse). Find the inverse of $f(x)=\frac{4x+3}{6x-4}$ for $x\ne \frac{2}{3}$.

$$y=\frac{4x+3}{6x-4}\Rightarrow 6xy-4y=4x+3\Rightarrow x(6y-4)=4y+3\Rightarrow x=\frac{4y+3}{6y-4}$$

So $f^{-1}(x)=\frac{4x+3}{6x-4}$ — this function is its own inverse. Check: $f(1)=\frac{7}{2}$ and $f\left(\frac{7}{2}\right)=\frac{17}{17}=1$.

Useful fact. $(g\circ f{)}^{-1}=f^{-1}\circ g^{-1}$.

An everyday example. Applying a 10 per cent discount and then a ₹50 coupon to a bill is a composite function — doing the two steps in the other order gives a different final amount.

The substance. Order matters in composition — $g\circ f$ means f acts first, even though g is written first.

**The graph of $f^{-1}$ is the reflection of the graph of f in the line $y=x$, because the point $(a,b)$ lies on the graph of f exactly when $(b,a)$ lies on the graph of $f^{-1}$.

Sketching steps:**

- Draw the line $y=x$
- Plot key points $(a,b)$ of f and mark the swapped points $(b,a)$
- Reflect the whole curve in $y=x$

Worked example. For $f(x)=2x+1$ the inverse is $f^{-1}(x)=\frac{x-1}{2}$.

- Points on f: $(0,1)$, $(1,3)$ and $(-1,-1)$
- Points on $f^{-1}$: $(1,0)$, $(3,1)$ and $(-1,-1)$
- The two graphs meet at $(-1,-1)$, which lies on $y=x$

Worked example 2. For $f(x)=x^3$, the inverse $\sqrt[3]{x}$ passes through $(8,2)$ because f passes through $(2,8)$.

An everyday example. Converting Celsius to Fahrenheit and back uses a function and its inverse, $F=\frac{9}{5}C+32$ and $C=\frac{5}{9}(F-32)$; for instance 100 °C gives 212 °F, and 212 °F returns 100 °C.

The substance. **For an increasing function, the graphs of f and $f^{-1}$ can meet only on the line $y=x$** — which is why solving $f(x)=x$ finds their meeting points.