Functions — IB DP Math AI SL

Question 1 • 2024 Exam Specimen 7 Marks

Question 2 • 2024 Exam Specimen 6 Marks

Question 3 • Exam Practice 5 Marks

Let f(x) = \sqrt{3x - 5} for x \geq \frac{5}{3} , and g(x) = x^2 + 1 for x \in \mathbb{R} .\n\n(a) Find an expression for (f \circ g)(x) . [2 marks]\n\n(b) Solve the equation (f \circ g)(x) = 5 . [3 marks]

Answer: (a) \ (f \circ g)(x) = \sqrt{3x^2 - 2}, \quad (b) \ x = 3 \text{ or } x = -3

Question 4 • Exam Practice 6 Marks

Let f(x) = \frac{3x - 5}{x - 2} for x \in \mathbb{R}, x \neq 2 . (a) Write down the equations of the vertical and horizontal asymptotes. [2 marks] (b) Find an expression for f^{-1}(x) and state its domain and range. [4 marks]

Answer: (a) \ x = 2, \ y = 3, \quad (b) \ f^{-1}(x) = \frac{2x - 5}{x - 3}, \ \text{Domain: } x \neq 3, \ \text{Range: } y \neq 2

Question 5 • Exam Practice 6 Marks

Consider the rational function f(x) = \frac{x^2 - 3x + 5}{x - 2} for x \ne 2 . (a) Write f(x) in the form ax + b + \frac{c}{x - 2} , where a, b, c \in \mathbb{Z} . [3 Marks] (b) Hence, write down the equations of the vertical asymptote and the oblique (slant) asymptote. [3 Marks]

Answer: (a) \ f(x) = x - 1 + \frac{3}{x - 2}, \qquad (b) \ \text{V.A.: } x = 2, \quad \text{Oblique: } y = x - 1

Question 6 • Exam Practice 6 Marks

The cubic polynomial P(x) = 2x^3 + ax^2 + bx - 6 has a factor of (x - 2) , and when divided by (x + 1) leaves a remainder of -18 . (a) Find the values of the constants a and b . [4 Marks] (b) Hence, factorize P(x) completely into linear factors over \mathbb{R} . [2 Marks]

Answer: (a) \ a = -5, \; b = 5, \qquad (b) \ P(x) = (x - 2)(2x^2 - x + 3)