- Define and explain Functions and graphs in your own words
- Use key terms such as Function accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
Pure mathematics studies structure for its own sake — and turns out to describe everything from planetary orbits to computer graphics. Here functions, sequences and calculus give you tools to model change precisely.
This lesson focuses on Functions and graphs: domain, range, composite and inverse functions, and transforming graphs of functions.
Domain, range, composite and inverse functions, and transforming graphs of functions.
Key ideas
Differentiating polynomials
From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule. So f(x) = 3x³ − 5x² + 2x − 7 gives f′(x) = 9x² − 10x + 2, and at x = 2 the gradient is 36 − 20 + 2 = 18.
Stationary points and nature
Solve f′(x) = 0 to find stationary points, then use the second derivative or a sign diagram: a positive second derivative means a minimum, a negative one a maximum. For f(x) = x³ − 3x, f′(x) = 3x² − 3 = 0 gives x = ±1, with a maximum at x = −1 and minimum at x = 1.
Key term — Function: A rule that assigns each input exactly one output, written y = f(x); the set of allowed inputs is the domain.
Given f(x) = 3x³ − 5x² + 2x − 7, find f′(x) and the gradient at x = 2.
Apply the power rule to each term: 3x³ → 9x², −5x² → −10x, 2x → 2, and the constant −7 differentiates to 0. So f′(x) = 9x² − 10x + 2. Substitute x = 2: 9(4) − 10(2) + 2 = 36 − 20 + 2 = 18.
Answer: f′(x) = 9x² − 10x + 2; gradient at x = 2 is 18.
- Differentiating the constant term to 1 or x The derivative of any constant is 0 — it contributes no change.
- Losing the constant of integration An indefinite integral must end with + C; different values of C give the whole family of antiderivatives.
Practice
dy/dx = 12x² + 6.
5 + 9 × 4 = 5 + 36 = 41.
2x³ + 4x + C.
[x²]₀³ = 9 − 0 = 9.
Quick check
Which of these best defines "Function"?
Find f(g(x)) given f(x) = x² and g(x) = 2x + 1.
Find the stationary points of y = x² − 6x + 5.
- Functions and graphs: domain, range, composite and inverse functions, and transforming graphs of functions.
- Differentiating polynomials: From first principles, the gradient of xⁿ is nxⁿ⁻¹ — the power rule.
- Derivative: The function f′(x) giving the gradient of y = f(x) at each point; the limit of the average rate of change as the interval shrinks to zero.
- Watch out for: differentiating the constant term to 1 or x