IB Questionbank

User interface language: English | Español

Date	November Example questions	Marks available	5	Reference code	EXN.1.SL.TZ0.8
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	8	Adapted from	N/A

Question

The following diagram shows the graph of $y = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

A function $f$ is defined by $f (x) = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

Describe a sequence of transformations that transforms the graph of $y = \sqrt{x}$ for $x \geq 0$ to the graph of $y = - 1 - \sqrt{x + 3}$ for $x \geq - 3$ .

[3]

State the range of $f$ .

[1]

Find an expression for $f^{- 1} (x)$ , stating its domain.

[5]

Find the coordinates of the point(s) where the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ intersect.

[5]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

for example,

a reflection in the $x$ -axis (in the line $y = 0$ ) A1

a horizontal translation (shift) $3$ units to the left A1

a vertical translation (shift) down by $1$ unit A1

Note: Award A1 for each correct transformation applied in a correct position in the sequence. Do not accept use of the “move” for a translation.

Note: Award A1A1A1 for a correct alternative sequence of transformations. For example,

a vertical translation (shift) down by $1$ unit, followed by a horizontal translation (shift) $3$ units to the left and then a reflection in the line $y = - 1$ .

[3 marks]

range is $f (x) \leq - 1$ A1

Note: Correct alternative notations include $] - \infty, - 1]$ , $(- \infty, - 1]$ or $y \leq - 1$ .

[1 mark]

$- 1 - \sqrt{y + 3} = x$ M1

Note: Award M1 for interchanging $x$ and $y$ (can be done at a later stage).

$\sqrt{y + 3} = - x - 1 (= - (x + 1))$ A1

$y + 3 = {(x + 1)}^{2}$ A1

so $f^{- 1} (x) = {(x + 1)}^{2} - 3 (f^{- 1} (x) = x^{2} + 2 x - 2)$ A1

domain is $x \leq - 1$ A1

Note: Correct alternative notations include $] - \infty, - 1]$ or $(- \infty, - 1]$ .

[5 marks]

the point of intersection lies on the line $y = x$

EITHER

${(x + 1)}^{2} - 3 = x$ M1

attempts to solve their quadratic equation M1

for example, $(x + 2) (x - 1) = 0$ or $x = \frac{- 1 \pm \sqrt{1^{2} - 4 (1) (- 2)}}{2} (x = \frac{- 1 \pm 3}{2})$

$- 1 - \sqrt{x + 3} = x$ M1

${(- 1 - \sqrt{x + 3})}^{2} = x^{2} \Rightarrow 2 \sqrt{x + 3} + x + 4 = x^{2}$

substitutes $2 \sqrt{x + 3} = - 2 (x + 1)$ to obtain $- 2 (x + 1) + x + 4 = x^{2}$

attempts to solve their quadratic equation M1

for example, $(x + 2) (x - 1) = 0$ or $x = \frac{- 1 \pm \sqrt{1^{2} - 4 (1) (- 2)}}{2} (x = \frac{- 1 \pm 3}{2})$

THEN

$x = - 2, 1$ A1

as $x \leq - 1$ , the only solution is $x = - 2$ R1

so the coordinates of the point of intersection are $(- 2, - 2)$ A1

Note: Award R0A1 if $(- 2, - 2)$ is stated without a valid reason given for rejecting $(1, 1)$ .

[5 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection

Show 83 related questions

22M.1.SL.TZ2.1a:
Find $g (0)$ .
22M.1.SL.TZ2.1c:
Find the value of $x$ such that $f (x) = 0$ .
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
21N.1.AHL.TZ0.2a.i:
the vertical asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2a.ii:
the horizontal asymptote of the graph of $f$ .
21N.1.AHL.TZ0.2c:
Sketch the graph of $f$ on the axes below.
18M.1.SL.TZ1.T_15b:
Write down the x-coordinate of P and the x-coordinate of Q.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
19M.2.AHL.TZ1.H_4a:
Write down the range of $f$ .
17M.1.AHL.TZ2.H_2a:
Write down the range of $f$ .
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.1.SL.TZ1.T_12b.ii:
Write down the number of solutions to $f(x) = - 6$ .
17M.2.AHL.TZ1.H_12c:
Explain why $f$ is an even function.
19N.1.SL.TZ0.S_10c:
The line ${L_2}$ is tangent to the graph of $g$ at ${\text{A}}$ and has equation $y = \left( {{\text{ln}}\,p} \right)x + q + 1$ .

The line ${L_2}$ passes through the point $\left( { - 2{\text{, }} - 2} \right)$ .

The gradient of the normal to $g$ at ${\text{A}}$ is $\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}$ .

Find the equation of ${L_1}$ in terms of $x$ .
18M.1.SL.TZ1.T_15c:
Write down the values of x for which $f\left( x \right) > g\left( x \right)$ .
21N.1.AHL.TZ0.2b.ii:
the $y$ -axis.
19M.2.SL.TZ1.T_4a:
Find the value of $f\left( x \right)$ when $x = \frac{1}{2}$ .
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
19N.1.SL.TZ0.S_10a:
Write down the coordinates of ${\text{B}}$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
19N.2.AHL.TZ0.H_3b:
Given that ${f^{ - 1}}\left( a \right) = 3$ , determine the value of $a$ .
17M.1.AHL.TZ2.H_2b:
Find an expression for ${f^{ - 1}}(x)$ .
EXN.1.SL.TZ0.8c:
Find an expression for $f^{- 1} (x)$ , stating its domain.
22M.1.AHL.TZ2.6b:
The range of $f$ is $a \leq y \leq b$ , where $a, b \in ℝ$ .

Find the value of $a$ and the value of $b$ .
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
22M.2.AHL.TZ1.10b.ii:
State the domain and range of $f^{- 1}$ .
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
19N.2.AHL.TZ0.H_3c:
Given that $g\left( x \right) = 2f\left( {x - 1} \right)$ , find the domain and range of $g$ .
20N.1.AHL.TZ0.H_10a.i:
Write down an expression for $f' (x)$ .
20N.1.AHL.TZ0.H_10b.i:
Show that $g^{- 1}$ exists.
20N.1.AHL.TZ0.H_12b:
State the equation of the horizontal asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12d:
Sketch the graph of $y = f (x)$ , stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.
20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
19N.2.AHL.TZ0.H_3a:
Find the value of $\left( {f \circ f} \right)\left( 1 \right)$ .
18M.1.SL.TZ1.T_15a:
Find the range of f.
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
21M.2.SL.TZ1.9e:
The line $L$ is tangent to the graphs of both $f$ and the inverse function $f^{- 1}$ .

Find the shaded area enclosed by the graphs of $f$ and $f^{- 1}$ and the line $L$ .
16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
20N.1.AHL.TZ0.H_10b.iii:
Hence find $g^{- 1} (x)$ .
20N.1.AHL.TZ0.H_10a.ii:
Hence, given that $f^{- 1}$ does not exist, show that $b^{2} - 3 a c > 0$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.1.SL.TZ1.T_12b.i:
Draw the line $y = - 6$ on the axes.
19N.1.SL.TZ0.S_10b:
Given that $f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}$ , find the equation of ${L_1}$ in terms of $x$ , $p$ and $q$ .
EXN.1.SL.TZ0.8b:
State the range of $f$ .
17M.2.AHL.TZ1.H_12e:
Find the inverse function ${g^{ - 1}}$ and state its domain.
21M.2.SL.TZ2.5a:
Find the range of $f$ .
17M.2.SL.TZ2.S_3c:
Write down the domain of $g$ .
21N.1.AHL.TZ0.2d:
The function $g$ is defined by $g (x) = \frac{a x + 4}{3 - x}$ , where $x \in ℝ, x \neq 3$ and $a \in ℝ$ .

Given that $g (x) = g^{- 1} (x)$ , determine the value of $a$ .
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.
16N.1.SL.TZ0.T_9b:
Find the value of $b$ .
16N.1.SL.TZ0.T_9c:
Write down the range of $f(x)$ .
17M.1.SL.TZ1.T_12c:
Find the range of values of $k$ for which $f(x) = k$ has no solution.
20N.1.AHL.TZ0.H_10b.ii:
$g (x)$ can be written in the form $p (x - 2)^{3} + q$ , where $p, q \in ℝ$ .

Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10d:
Sketch the graphs of $y = g (x)$ and $y = g^{- 1} (x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.
20N.1.AHL.TZ0.H_12a:
State the equation of the vertical asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12c:
Use an algebraic method to determine whether $f$ is a self-inverse function.
20N.1.AHL.TZ0.H_12e:
The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .
17M.2.AHL.TZ1.H_12g.ii:
Hence, show that there are no solutions to $({g^{ - 1}})'(x) = 0$ .
17N.2.AHL.TZ0.H_10d:
This region is now rotated through $2\pi$ radians about the $x$ -axis. Find the volume of revolution.
19M.2.AHL.TZ1.H_4b:
Find $f'\left( x \right)$ , stating its domain.
21M.2.AHL.TZ2.12e:
State the domain of $g^{- 1}$ .
20N.1.AHL.TZ0.H_10c:
State each of the transformations in the order in which they are applied.
17N.1.SL.TZ0.S_3c:
On the grid, sketch the graph of ${f^{ - 1}}$ .
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
17M.1.AHL.TZ2.H_2c:
Write down the domain and range of ${f^{ - 1}}$ .
17M.2.SL.TZ1.S_10a.iii:
Write down the value of $k$ .
16N.1.SL.TZ0.T_9a:
Write down the equation of the axis of symmetry for this graph.
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.2.SL.TZ1.T_4c:
Find the coordinates of the bridges relative to the centre of Orangeton.
20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of $34.50$ represents.
19M.2.SL.TZ1.T_4e:
This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.
20N.1.SL.TZ0.T_2c:
Kaelani has $450 PGK$ .

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
19M.2.SL.TZ1.T_4b.ii:
State the domain of $P$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
19M.2.SL.TZ1.T_4b.i:
Find the function, $P\left( x \right)$ , that would define this footpath on the map.

Hide related questions

Topic 2—Functions » SL 2.5—Composite functions, identity, finding inverse

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options