Composite and Inverse of Functions Worked ExamplesIGCSE Maths 0580

$fg$ means $f(g(x))$: the function written nearest to $x$ acts first. Reading left to right reverses the whole calculation. A composite is not a product either: the June 2025 examiner report noted that "The main error at the start was to treat fg(x) as a product of two functions" — here that would give $5 \times 4 = 20$ for both parts.

Substituting at $x = 2$ checks both: $3(4) - 1 = 11$ ✓ and $9(4) - 12 + 1 = 25$ ✓, matching the previous example. $(3x-1)^2$ is not $9x^2 + 1$. Substitute one function into the other; do not multiply them. The June 2025 examiner report noted that "Common incorrect answers involved attempting to multiply the two functions."

$f^{-1}$ means the inverse function, not $\frac{1}{f(x)}$. The March 2026 examiner report found that "Some candidates found the reciprocal of the function rather than the inverse function." The March 2025 report also noted "errors including leaving the answer in terms of y or incorrect signs when rearranging", so finish with $f^{-1}(x) = \ldots$ in terms of $x$. The verification in (b) costs two lines and catches any rearrangement error, so it is worth doing even when not asked.

Same routine as making a subject when it appears twice: clear fractions, expand, gather, factorise, divide. Check at $x = 3$: $h(3) = 3$ and $h^{-1}(3) = 3$ — a fixed point, so try another, $h(2) = 4$ and $h^{-1}(4) = 2$. ✓

The shortcut in (b) is worth knowing — it turns a rearrangement into a single substitution. On a question of this type the June 2025 examiner report said "Most candidates struggled with this part. The best method began with x = h(2)." — the same idea as writing $x = f(4)$ here. It works because $f$ and $f^{-1}$ undo one another, so reading the statement backwards gives the answer immediately.

$ff(x)$ is sometimes written $f^2(x)$, and it does not mean $\big(f(x)\big)^2$. Here $ff(x) = 9x - 4$ while $\big(f(x)\big)^2 = 9x^2 - 6x + 1$ — two quite different functions.

An inverse undoes the last operation first — like taking off your shoes before your socks. In (a) the whole of $2x + 3$ is divided by $5$, so the fraction line must cover both terms; $2x + \frac{3}{5}$ loses the mark. A quick test value, as in (c) and (d), catches most errors.

In $fg(3)$ the function written next to the $3$ acts first, so read $g$ before $f$. For an inverse, read the graph backwards, from the output on the $y$-axis to the input on the $x$-axis. Part (d) has two layers: stopping at $u = 0$ and $u = 4$ answers the wrong question.

The $\pm$ from the square root must be settled using the domain — the negative root gives the reflection of the dashed half instead. Solving $f(x) = x$ is valid here because $f$ is increasing on its domain, and an increasing function can only meet its inverse on the line $y = x$. Domain and range simply swap between $f$ and $f^{-1}$.

Negatives cause most of the lost marks. On a question like part (b), the March 2025 examiner report found that dealing with the negative values led to errors, and a common one was not changing the sign of the second term when multiplying out the bracket. The same slip here gives $9 - 10 - 6x = -1 - 6x$. In (c), $f^{-1}$ is the inverse, not a power: the June 2025 examiner report noted candidates who "misinterpreted the notation as a power of negative 1 rather than an inverse function".

Turn every inverse statement back into a statement about $h$. The June 2025 examiner report said this type of part "was found to be challenging", and that "Some candidates attempted to use logs to answer the question, but these were often applied incorrectly." Logs are not needed: $h^{-1}(x) = 3$ simply means $x = h(3)$. In (c) do not work out $h(8) = 4^8$ — on a similar question the March 2026 examiner report noted that "Many made the common error of calculating h(10 000)."

Order of operations matters: square before multiplying by $2$. The March 2026 examiner report noted that "Some candidates multiplied the terms in the bracket by 2 before squaring." Here that gives $(8 - 6x)^2 - 5 = 36x^2 - 96x + 59$, which is wrong. A quick numerical check, as in step 4, catches this.

The June 2025 examiner report noted that "The main error at the start was to treat fg(x) as a product of two functions". Multiplying gives $8x^2 + 14x - 15$, which is not of the form $ax + b$ — a sign that something has gone wrong. Notice that $fg(x)$ and $gf(x)$ are different here, so work out the composite each part asks for rather than reusing one answer.