Pure: Algebra and Functions
Algebra and functions covers manipulating expressions, solving equations and inequalities, and analysing functions such as their domain, range, composition and inverse. It includes surds, indices, quadratics, algebraic fractions, the factor and remainder theorems, and graph transformations, and underpins almost every other Pure Maths topic, from coordinate geometry to calculus.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Simplify surds and indices first, using rules like a^m x a^n = a^(m+n), and rationalise denominators by multiplying by a suitable conjugate.
- Complete the square or factorise quadratics to solve equations, sketch graphs or prove inequalities.
- For algebraic fractions, factorise numerator and denominator fully before cancelling common factors.
- For functions, work out domain and range from the algebraic form or graph, and find composite functions fg(x) by substituting g(x) into f.
- Find an inverse function by writing y = f(x), making x the subject, then rewriting as f^-1(x); state its domain from the range of f.
- For inequalities and modulus equations, reason about the critical values, then check which regions or cases satisfy the original equation or inequality.
Worked example
The function f is defined by f(x) = 3/(x - 2), x is real, x not equal to 2. Find f^-1(x), stating its domain.
- Write y = 3/(x - 2).
- Multiply both sides by (x - 2): y(x - 2) = 3.
- Expand: yx - 2y = 3.
- Rearrange to make x the subject: x = (3 + 2y)/y = 3/y + 2.
- Swap to function notation: f^-1(x) = 3/x + 2.
- Final answer: f^-1(x) = 3/x + 2, with domain x not equal to 0 (since 0 is excluded from the range of f).
Practice questions
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Q1Simplify fully (4x^6)^(1/2).Show answer
Answer: 2x^3 (square root 4 gives 2, halve the power of x^6).
Q2Express 6/(sqrt(7) - 2) in the form a + b*sqrt(7), where a and b are integers.Show answer
Answer: 4 + 2sqrt(7) (a = 4, b = 2), found by multiplying top and bottom by (sqrt(7) + 2).
Q3Solve the inequality 3x^2 - 5x - 2 <= 0.Show answer
Answer: -1/3 <= x <= 2 (factorise as (3x+1)(x-2), critical values -1/3 and 2).
Q4Simplify (x^2 - 16)/(x^2 - x - 12), stating any values of x for which it is undefined.Show answer
Answer: (x + 4)/(x + 3), x not equal to 4 or -3.
Q5Functions f and g are defined by f(x) = x^2 - 3 and g(x) = 2x + 1. Find fg(2) and gf(2).Show answer
Answer: fg(2) = 22 and gf(2) = 3 (apply the inner function first each time).
Q6Solve the equation |3x - 4| = x + 2.Show answer
Answer: x = 3 or x = 0.5 (solve 3x-4=x+2 and 3x-4=-(x+2), checking both satisfy the original equation).
Exam-style questions
Written in the style of a A Level Maths exam paper, with a full mark scheme.
Express f(x) = x^2 - 10x + 32 in the form (x - p)^2 + q, stating the values of p and q, and hence state the coordinates of the minimum point of the graph of y = f(x).
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f(x) = 2x^3 + 3x^2 - 11x - 6. Use the factor theorem to show that (x + 3) is a factor of f(x), and hence factorise f(x) completely.
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A line l has equation y = 2x - 2, and a curve C has equation y = x^2 - 4x + 7. Show that l is a tangent to C, and find the coordinates of the point where they touch.
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Free printable worksheet
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