IB Maths AA SL Topic 2: Functions questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

Easy Medium Hard

Question 1

Paper 1, no calculatorMedium6 marks

The function $f$ is defined by

f(x)=(x-2)^2+1,\qquad x\geq 2.

(a)

Explain why $f$ has an inverse function.

[1]

(b)

Find $f^{-1}(x)$ stating its domain.

[3]

(c)

Using your answer to (b), determine $(f^{-1}\circ f)(5)$ showing the value of $f(5)$

[2]

Question 2

Paper 2, calculatorMedium7 marks

A quadratic function $f$ has $x$ -intercepts $(-2,0)$ and $(6,0)$ Its graph passes through $(0,24)$

(a)

Determine $f(x)$ in factored form.

[3]

(b)

Hence write $f(x)$ in vertex form and state the equation of its axis of symmetry.

[4]

Question 3

Paper 1, no calculatorMedium7 marks

For real $k$ let $q_k(x)=x^2-4x+k$ where $x\in\mathbb R$

(a)

Solve $q_3(x)=0$

[2]

(b)

Hence solve $q_3(x)\leq 0$

[2]

(c)

Determine the values of $k$ for which $q_k(x)=0$ has: (i) two distinct real roots; (ii) two equal real roots; (iii) no real roots.

[3]

Question 4

Paper 2, calculatorMedium7 marks

For real $t$ let $q_t$ be the quadratic function defined by

q_t(x)=x^2+(t-4)x+2t+1.

(a)

Determine the discriminant, $\Delta_t$ of $q_t(x)$

[2]

(b)

Given that $0\leq t\leq 10$ determine the set of values of $t$ for which $q_t(x)=0$ has two distinct real roots.

[3]

(c)

Hence, identify the integer value of $t$ in this set and solve $q_t(x)\leq 0$ for $x\in\mathbb{R}$

[2]

Question 5

Paper 2, calculatorMedium6 marks

For real $k$ consider the equation

(k+1)x^2-4x+1=0.

(a)

Determine the values of $k$ for which the equation has two distinct real roots, two equal real roots, or no real roots. State separately what happens when the equation is not quadratic.

[3]

(b)

Hence, when $k=2$ solve the inequality

(k+1)x^2-4x+1\leq 0.

[3]

Question 6

Paper 2, calculatorMedium5 marks

Let $f(x)=\dfrac{1}{x}$ $x\ne 0$ and define

g(x)=3+2f(x-1)=\frac{3x-1}{x-1},\qquad x\ne 1.

(a)

State the inverse function of $f$

[1]

(b)

Hence determine $g^{-1}(x)$ and state the equations of its vertical and horizontal asymptotes.

[4]

Question 7

Paper 1, no calculatorMedium5 marks

The graph of the function $r$ is shown in Figure 1. It is known that

r(x)=\frac{ax+b}{x+d}, \qquad x\ne-d,

where $a,b,d\in\mathbb R$

(a)

Determine the values of $a$ $b$ and $d$

[3]

(b)

Hence, show that $r$ is self-inverse, stating the domain of $r^{-1}$

[2]

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