IGCSE Maths Functions & differentiation topical questions

f(x) = 3x − 2. (Extended)

1. (a)Find f(4).[1]
2. (b)Find f(−1).[1]
3. (c)Find f⁻¹(x).[2]

f(x) = x² and g(x) = 2x + 3. (Extended)

1. (a)Find gf(2).[2]
2. (b)Find fg(2).[2]
3. (c)Solve gf(x) = 21.[1]

f(x) = 5x + 4. (Extended)

1. (a)Find f(3).[1]
2. (b)Find f⁻¹(x).[3]
3. (c)Use your answer to (b) to verify that f⁻¹(f(3)) = 3.[1]

Differentiate each expression with respect to x. (Extended)

1. (a)7x²[1]
2. (b)x⁴ − 3x² + 5x[2]
3. (c)8⁄x[1]

The curve is y = x² − 6x + 2. (Extended)

1. (a)Find dy⁄dx.[1]
2. (b)Find the coordinates of the stationary point.[2]
3. (c)Use the second derivative to show that the point is a minimum.[2]

f(x) = 2(x + 5). (Extended)

1. (a)Find f⁻¹(x).[3]
2. (b)Solve f(x) = 40.[1]
3. (c)Describe how the graphs of f and f⁻¹ are related.[1]

f(x) = x² − 9 and g(x) = 2x. (Extended)

1. (a)Find fg(3).[2]
2. (b)Find gf(3).[2]
3. (c)Solve gf(x) = 0.[2]

Differentiate each expression with respect to x. (Extended)

1. (a)x⁴ − 3x[2]
2. (b)3⁄x²[2]
3. (c)x√x[2]

The curve is y = x³ − 12x + 1. (Extended)

1. (a)Find dy⁄dx.[1]
2. (b)Find the coordinates of both stationary points.[2]
3. (c)Classify each stationary point.[2]

The curve is y = x². (Extended)

1. (a)Find the gradient of the tangent at x = 3.[2]
2. (b)Find the equation of the tangent at that point.[2]
3. (c)Write down the gradient of the normal at x = 3.[1]

f(x) = 2x − 4 and g(x) = 1⁄(x + 1). (Extended)

1. (a)State the domain of g.[1]
2. (b)Find f⁻¹(x).[2]
3. (c)Find and simplify gf(x).[2]
4. (d)Solve gf(x) = 1.[1]

The curve is y = 2x³ + 3x² − 36x + 7. (Extended)

1. (a)Find dy⁄dx.[2]
2. (b)Find the two x-coordinates of the stationary points.[2]
3. (c)Use the second derivative to classify each stationary point.[2]

The curve is y = x³ − 6x² + 11x − 6. (Extended)

1. (a)Show that the gradient of the curve at the point (1, 0) is 2.[2]
2. (b)Find the equation of the tangent at the point (1, 0), in the form y = mx + c.[2]
3. (c)The tangent meets the curve again at another point P. Given that x³ − 6x² + 9x − 4 = (x − 1)²(x − 4), find the coordinates of P.[3]

f(x) = (x − 1)² for x ⩾ 1. (Extended)

1. (a)Write down the range of f, with a reason.[2]
2. (b)Find f⁻¹(x).[3]
3. (c)State the domain of f⁻¹.[1]