Revise Functions: domain, range, composite and inverse for IB Mathematics: Analysis and Approaches (2.2)

A function $f$ is a rule that maps each input in its domain to exactly one output. We write $f(x)$ for the output when the input is $x$, and $f:x\mapsto f(x)$ for the mapping itself. The single-output requirement is the whole point: a rule that could send one input to two different outputs is not a function. Functions are then classified by how outputs are shared. A function is one-to-one if every output comes from exactly one input, and many-to-one if some output is produced by two or more different inputs. This distinction decides later whether an inverse can exist.

The domain is the complete set of permitted inputs; the range is the complete set of outputs the function actually produces. When finding a domain in SL Maths, watch for the two standard restrictions: you may never divide by zero, and you may never take an even root (such as a square root) of a negative number. Finding a range is about what the rule does to its inputs — squaring can never give a negative, a reciprocal can never give zero — so look for maxima, minima and asymptotes.

A composite function is formed when the output of one function becomes the input of another. The notation $(f\circ g)(x)$ reads 'f composed with g of x' and means: apply $g$ first, then feed its result into $f$. You build it by substituting the whole expression for $g(x)$ into every $x$ of $f(x)$, working from the inside out.

(f\circ g)(x) = f(g(x))

An inverse function $f^{-1}$ undoes the action of $f$: if $f$ maps $a$ to $b$, then $f^{-1}$ maps $b$ back to $a$. For an inverse to exist over the whole domain, $f$ must be one-to-one — otherwise a single output would have to map back to two different inputs, which no function is allowed to do. You can test this graphically with the horizontal line test. To find the inverse algebraically, write $y=f(x)$, swap $x$ and $y$, and solve for $y$; swapping the variables is precisely what reverses input and output.

In Paper 1 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach — a valid substitution or a sound rearrangement — even if the arithmetic later slips. An accuracy mark (A) rewards a correct result and is dependent on the corresponding method being present. IB conventions also let follow-through (FT) protect a later mark that is correctly worked from an earlier (possibly wrong) value, allow 'ignore subsequent working' (ISW) once a correct answer is seen, award an answer that is given (AG) only when the full derivation is shown, and accept any equivalent correct form. Study how each mark below is earned by one specific line.

Functions are the backbone of the rest of the course. The domain-and-range skills reappear the moment you meet rational, exponential and logarithmic functions, each with its own restrictions and asymptotes. Composition underlies the chain rule in calculus, where differentiating $f(g(x))$ is exactly unpicking a composite. Inverses return with logarithms as the inverse of exponentials and with inverse trigonometric functions, and the reflection in $y=x$ is the graphical fingerprint of every one of them. Master the habit — classify the function, state the domain and range in set notation, respect the order of composition, and check one-to-one before inverting — and the functions work throughout the course becomes variations on a method you already own.