Are functions on the Foundation maths paper?

Only in part. Spec ref A7 asks candidates at both tiers to read a simple expression as a function with an input and an output, usually through function machines. The inverse function, the composite function and the formal f(x) notation that goes with them are Higher tier only, so a Foundation paper will not ask for fg(x) or f^-1(x).

Does fg(x) mean do f first or g first?

Do g first. fg(x) is shorthand for f(g(x)), so the function nearest the bracket goes first and its output becomes the input of the other one. The two orders normally give different functions: with f(x) = 2x + 3 and g(x) = x^2, fg(4) = 35 but gf(4) = 121.

How many marks are functions worth on Edexcel Higher?

Usually 4-6 marks, set as one question with two or three short parts on a single paper. The parts are cheap individually, often 1 mark for evaluating f(10) or naming the excluded value and 2-3 marks for an inverse or a composite, which makes it one of the quickest algebra topics to bank.

Is f inverse the same as 1 over f(x)?

No. The -1 in f^-1(x) is notation for the inverse function, the rule that undoes f, not an index. For f(x) = 2x - 1 the inverse is f^-1(x) = (x + 1)/(2), whereas (1)/(f(x)) = (1)/(2x - 1). Substituting a value into both shows they are different: f^-1(5) = 3 but (1)/(f(5)) = (1)/(9).

Functions - Edexcel GCSE Higher Maths

Typically 4-6 marks in a series, usually one multi-part question · Spec 1MA1

Function notation names a rule and its input. Writing $f(x) = 3x + 4$ says the function called $f$ multiplies whatever goes in by 3 and then adds 4, so $f(5) = 19$ . On Edexcel's 1MA1 specification this is spec ref A7. Reading a simple expression as a function with an input and an output is common to both tiers, but the two ideas that carry the marks, the reverse process written as the inverse function $f^{-1}(x)$ and the succession of two functions written as the composite function $fg(x)$ , are Higher tier only. Foundation candidates work with function machines and input-output tables and never meet $fg(x)$ .

Edexcel almost always sets this as a single question split into short parts, 4-6 marks in total, and the parts repeat from series to series: work out $f(10)$ for 1 mark, find $g^{-1}(x)$ for 2 marks, show that $ff(x) = 9x - 48$ when $f(x) = 3(x - 4)$ for 2 marks, or state the value of $x$ that must be excluded from any domain of $g$ for 1 mark. It can land on any of the three papers. A calculator buys you nothing here, since every part is algebra, and when a stem says show clear algebraic working an answer sitting on its own scores nothing even when it is right.

The marks go missing in the same few places. The order in $fg(x)$ runs right to left, $g$ first, so a candidate who works left to right is wrong from the opening line and no method mark survives. The $-1$ in $f^{-1}(x)$ is notation for the inverse, not a power, so it is not $\frac{1}{f(x)}$ . And an inverse that stops at $x = \frac{y - 5}{2}$ has stopped a line early: the answer has to come back in $x$ , written as $f^{-1}(x)$ . Drill the swap-and-rearrange routine until it is automatic, and read $fg$ aloud as f of g of x so the order fixes itself.

Worked example

The functions

f

and

g

are such that

f(x) = 2x + 5

g(x) = \frac{6}{x - 4}

(a) Work out

fg(7)

. (2 marks) (b) Find

f^{-1}(x)

. (2 marks) (c) State the value of

x

that must be excluded from any domain of

g

. (1 mark)

[5 marks]

Work from the inside out. The inner function is $g$ , so $g(7) = \frac{6}{7 - 4} = \frac{6}{3} = 2$ .
M1: substituting 7 into $g$ first, not into $f$
That output is now the input of $f$ : $f(2) = 2(2) + 5 = 9$ , so $fg(7) = 9$ .
A1: 9 (a candidate who wrote $f(7) = 19$ first has already lost both marks)
For the inverse, write $y = 2x + 5$ and swap the letters: $x = 2y + 5$ .
M1: a correct start, either swapping the variables or making $x$ the subject
Rearrange for $y$ : $2y = x - 5$ , so $f^{-1}(x) = \frac{x - 5}{2}$ .
A1: correct inverse given in terms of $x$ ; an answer left as $x = \frac{y - 5}{2}$ does not get this mark
The denominator of $g$ is $x - 4$ , and $x - 4 = 0$ when $x = 4$ . Dividing by zero has no value, so $x = 4$ is excluded.
B1: 4, from setting the denominator equal to zero

Practice questions

These are original questions written in Edexcel 1MA1 Higher style, running from single substitutions up to composite functions, inverses and the input a denominator rules out. Functions can appear on any of the three papers and a calculator adds nothing, so work them by hand as you would on Paper 1. Where a question asks for an expression, give it in terms of $x$ ; equivalent forms are accepted, so (x-1)/4 and 0.25x - 0.25 both mark correct.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

Question 1 Warm-up [1 mark]

$f(x) = \frac{6}{x - 3}$ Work out the value of $f(5)$ .

Your answer (fractions like 1/2 are fine)
Question 2 Warm-up [2 marks]

$f(x) = x^2 + 2$ and $g(x) = x - 5$ Work out the value of $fg(9)$ .

Your answer (fractions like 1/2 are fine)
Question 3 Exam pace [2 marks]

A taxi firm charges a fixed £2.50 plus £1.80 for each kilometre travelled. The cost in pounds of a journey of $x$ km is $f(x) = 2.5 + 1.8x$ . Work out $f(6)$ , the cost in pounds of a 6 km journey.

Your answer (fractions like 1/2 are fine)
Question 4 Exam pace [2 marks]

$f(x) = x^2 + 1$ and $g(x) = x - 4$ Work out the value of $fg(3)$ .

Your answer (fractions like 1/2 are fine)
Question 5 Stretch [2 marks]

$f(x) = \frac{10}{x}$ , defined for every value of $x$ apart from 0. Work out the value of $ff(6)$ .

Your answer (fractions like 1/2 are fine)
Question 6 Stretch [3 marks]

$f(x) = 4x - 3$ Given that $f^{-1}(k) = 5$ , work out the value of $k$ .

Your answer (fractions like 1/2 are fine)

Common mistakes examiners see

Working $fg(x)$ from left to right. With $f(x) = 2x + 3$ and $g(x) = x^2$ , that turns $fg(4)$ into $g(11) = 121$ when the answer is $f(16) = 35$ .

Rewrite $fg(x)$ as $f(g(x))$ before you calculate anything. The letter next to the bracket is applied first, so the number goes into $g$ , and the output of $g$ becomes the input of $f$ .
Reading the $-1$ as a power and writing $f^{-1}(x) = \frac{1}{3x + 1}$ . That is the reciprocal of $f$ , a different function entirely.

Use the same three lines every time: $y = 3x + 1$ , swap to $x = 3y + 1$ , make $y$ the subject, giving $f^{-1}(x) = \frac{x - 1}{3}$ . Check it by reversing a known value: $f(2) = 7$ , so $f^{-1}(7)$ must come back as 2.
Treating $fg(x)$ as $f(x) \times g(x)$ . With $f(x) = x + 1$ and $g(x) = 2x - 3$ that produces $(x + 1)(2x - 3)$ instead of $2x - 2$ .

There is no multiplication sign in $fg(x)$ because nothing is multiplied. Substitute the whole of $g(x)$ into $f$ wherever an $x$ appears in $f$ , brackets and all.
Answering solve $f(x) = 10$ by evaluating $f(10)$ . For $f(x) = 3x - 2$ that hands in 28 when the answer is $x = 4$ .

Check which side the number is on. $f(10)$ is an instruction to substitute; $f(x) = 10$ is an equation, so write $3x - 2 = 10$ and solve it for $x$ .
Giving the excluded value with the wrong sign: for $g(x) = \frac{6}{x - 4}$ , answering $x = -4$ because the 4 is subtracted.

Set the denominator equal to zero and solve it like any other equation. $x - 4 = 0$ gives $x = 4$ , and that single input is the one the function cannot accept.

Frequently asked questions

Are functions on the Foundation maths paper?: Only in part. Spec ref A7 asks candidates at both tiers to read a simple expression as a function with an input and an output, usually through function machines. The inverse function, the composite function and the formal $f(x)$ notation that goes with them are Higher tier only, so a Foundation paper will not ask for $fg(x)$ or $f^{-1}(x)$ .
Does fg(x) mean do f first or g first?: Do $g$ first. $fg(x)$ is shorthand for $f(g(x))$ , so the function nearest the bracket goes first and its output becomes the input of the other one. The two orders normally give different functions: with $f(x) = 2x + 3$ and $g(x) = x^2$ , $fg(4) = 35$ but $gf(4) = 121$ .
How many marks are functions worth on Edexcel Higher?: Usually 4-6 marks, set as one question with two or three short parts on a single paper. The parts are cheap individually, often 1 mark for evaluating $f(10)$ or naming the excluded value and 2-3 marks for an inverse or a composite, which makes it one of the quickest algebra topics to bank.
Is f inverse the same as 1 over f(x)?: No. The $-1$ in $f^{-1}(x)$ is notation for the inverse function, the rule that undoes $f$ , not an index. For $f(x) = 2x - 1$ the inverse is $f^{-1}(x) = \frac{x + 1}{2}$ , whereas $\frac{1}{f(x)} = \frac{1}{2x - 1}$ . Substituting a value into both shows they are different: $f^{-1}(5) = 3$ but $\frac{1}{f(5)} = \frac{1}{9}$ .

Functions - Edexcel GCSE Higher Maths

Worked example

Practice questions

Common mistakes examiners see

Frequently asked questions

Related topics