Revise Functions: domain, range, inverse and graphs for IB Mathematics: Applications and Interpretation (2.2)

A function is a rule that assigns to each input exactly one output. We write $f(x)$, read as '$f$ of $x$', where $f$ names the rule and $x$ is the input; the whole symbol $f(x)$ is the output. So $f(2)$ means 'put $2$ into the rule' — for $f(x)=3x-2$ it gives $f(2)=3(2)-2=4$. The one rule that must never be broken is that a single input cannot lead to two different outputs. Graphically this is the vertical line test: if any vertical line meets the graph more than once, that $x$ would have two $y$-values, so the graph is not a function.

The domain is the set of allowed inputs — every $x$ you may feed the rule. The range is the set of outputs — every $y$ the rule actually produces over that domain. For $f(x)=x^2$ the domain can be all real numbers, $x \in \mathbb{R}$, but the range is only $y \ge 0$, because squaring never gives a negative. Two things can limit a domain. The algebra forbids some inputs — you cannot divide by zero, so $\frac{1}{x-4}$ excludes $x=4$, and you cannot square-root a negative, so $\sqrt{x-1}$ needs $x \ge 1$. A real context forbids others — a length, a time after an event or a number of items cannot be negative.

Reading the range from a graph is a genuine exam skill, not a formula. Find the turning points and the behaviour at the ends. A parabola with a minimum at $(2,-3)$ that opens upward reaches every height at or above $-3$, so its range is $y \ge -3$. If the domain is restricted, the range shrinks with it: the same parabola on $x \ge 2$ (its right half only) still has range $y \ge -3$, but on $0 \le x \le 2$ it runs from the minimum up to $f(0)$.

The inverse of $f$, written $f^{-1}$, runs the process backwards. If $f$ takes $a$ to $b$ — that is $f(a)=b$ — then $f^{-1}$ takes $b$ back to $a$, so $f^{-1}(b)=a$. Because inputs and outputs trade places, the domain of $f^{-1}$ is the range of $f$ and the range of $f^{-1}$ is the domain of $f$. Not every function can be reversed cleanly: it must be one-to-one, meaning each output comes from a single input, which you can check with the horizontal line test. To build the inverse algebraically, undo the rule step by step by swapping $x$ and $y$ and solving.

To find the inverse of $f(x)$ algebraically: \n 1. Write the function as $y = f(x)$. \n 2. Swap the variables $x$ and $y$. \n 3. Rearrange to make $y$ the subject. \n 4. Replace $y$ with $f^{-1}(x)$.

There is a clean geometric picture behind all of this: the graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$. It follows directly from the coordinate swap — every point $(a,b)$ on the original becomes $(b,a)$ on the inverse, and reflecting in $y=x$ is precisely the transformation that interchanges the two coordinates. That is why the two graphs always meet the line $y=x$ symmetrically, and it gives you a quick way to sketch an inverse without any algebra.

On Paper 2 the calculator is the tool of choice for graphs. Enter a function to see its shape, then use the built-in features to read off what a question asks for. The 'zero' (or root) feature gives the $x$-intercepts — where $f(x)=0$; evaluating $f(0)$ gives the $y$-intercept. The 'maximum' and 'minimum' features locate turning points, which is exactly how you find the ends of a range. And the 'intersect' feature finds where two graphs meet — the graphical way to solve $f(x)=g(x)$. In each case, quote the coordinates the GDC returns to the accuracy the question requires, usually 3 significant figures.

In IB Mathematics the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach; an accuracy mark (A) rewards a correct result and depends on the method mark it follows, so an A-mark cannot be earned without its M-mark. Follow-through (FT) means an earlier slip need not cost the marks that depend on it, provided the later step is carried out correctly on your own figures. On Paper 2, GDC graphing is expected, but an instruction to find $f^{-1}(x)$ still requires the algebraic swap-and-solve. Study how each mark below is earned by a specific line — that is exactly how the Practice button will mark your own attempt.

Domain, range and inverse are the grammar of every function you will meet next. Reading a range from a graph is exactly the skill you reuse for the outputs of exponential, logarithmic and trigonometric models; the reflection in $y=x$ reappears when the logarithm is introduced as the inverse of the exponential; and the discipline of choosing a sensible domain is what makes a real-world model trustworthy. Above all, the habit of showing method — write the rule, swap, solve, and quote the GDC to the stated accuracy — is the discipline every function question on both papers rewards. Get the three headlines right — allowed inputs, produced outputs, and the process reversed — and the rest of the functions strand becomes variations on ideas you already control.