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

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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 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 3

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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