IB Mathematics Analysis and Approaches Topic SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 2.2—functions, Notation Domain, Range and Inverse as Reflection with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 2.2—functions, Notation Domain, Range and Inverse as Reflection and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 2

A function $f$ is defined by $f (x) = cos^{- 1} (\frac{1}{x ^{2} + 1})$ for $x \in R$ . A function $g$ is defined by $g (x) = cos^{- 1} (\frac{1}{x ^{2} + 1})$ for $x \geq 0$ .

State the domain $D$ of $f$ .

[2]

Sketch the graph of $y = f (x)$ , showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

[3]

Explain why $f$ is an even function.

[2]

State why the inverse function $f^{- 1}$ does not exist.

[1]

Let $g$ be the restriction of $f$ to $x \geq 0$ . Find the inverse function $g^{- 1} (x)$ and state its domain.

[4]

Question 2

SLPaper 2

Let $f(x)=e^{kx},\quad g(x)=\ln(x+2),\quad x>-2,\ k>0.$ It is given that $(f \circ g) (3) = 4$ .

Determine $k$ correct to 3 significant figures.

[3]

Find the composites and their maximal domains.

[4]

Solve $(x + 2)^{k} = ln (2 + e^{k x})$ for $x \geq 0$ (3 d.p.) and justify uniqueness.

[7]

Find $(f \circ g)^{- 1} (y)$ with domain and range.

[4]

Question 3

SLPaper 2

Let $p \in R$ and $q_{p} (x) = x^{2} - (p + 8) x + (2 p + 8)$ .

Find all real values of $p$ for which $q_{p}$ has two distinct real roots and both roots are greater than $1$ .

[6]

Let $p = 3$ (not necessarily satisfying the condition in the previous part) so that $q (x) = x^{2} - 11 x + 14$ , and $g (x) = ln (x + 2) + 1$ . Find the exact roots of $q (x) = 0$ .

[3]

Solve $q (x) = g (x)$ for $x \geq 0$ , giving solutions correct to $3$ d.p., and justify the number of solutions.

[6]

Question 4

SLPaper 2

For a real parameter $p$ , consider the quadratic $q_p(x)=x^2-(p+2)x+(2p-5).$

Find all real values of $p$ for which $q_{p}$ has two distinct real roots and both roots are greater than $0$ .

[6]

Let $p = 4$ so that $q (x) = x^{2} - 6 x + 3$ , and let $g (x) = ln (x + 2) + 1$ (domain $x > - 2$ ).

Find the exact roots of $q (x) = 0$ .

[3]

Solve $q (x) = g (x)$ for $x \geq 0$ to three decimal places. Justify the number of solutions on $x \geq 0$ .

[6]

Question 5

SLPaper 2

Consider the function $f (x) = \frac{1}{x + 3}$ .

Find the domain for the function $f (x)$ .

[2]

Find the range for the function $f (x)$ .

[2]

Sketch the graph of the function $f (x)$ .

[4]

IB Mathematics Analysis and Approaches SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank

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IB Mathematics Analysis and Approaches SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 2.2—functions, Notation Domain, Range and Inverse as Reflection Questionbank

Unlock all 18 questions with a free account