IB Questionbank

User interface language: English | Español

Date	May 2022	Marks available	2	Reference code	22M.1.SL.TZ1.4
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Find	Question number	4	Adapted from	N/A

Question

Consider the functions $f (x) = \sqrt{3} \sin x + \cos x$ where $0 \leq x \leq π$ and $g (x) = 2 x$ where $x \in ℝ$ .

Find $(f \circ g) (x)$ .

[2]

Solve the equation $(f \circ g) (x) = 2 \cos 2 x$ where $0 \leq x \leq π$ .

[5]

Markscheme

$(f \circ g) (x) = f (2 x)$ (A1)

$f (2 x) = \sqrt{3} \sin 2 x + \cos 2 x$ A1

[2 marks]

$\sqrt{3} \sin 2 x + \cos 2 x = 2 \cos 2 x$

$\sqrt{3} \sin 2 x = \cos 2 x$

recognising to use $\tan$ or $cot$ M1

$\tan 2 x = \frac{1}{\sqrt{3}}$ OR $cot 2 x = \sqrt{3}$ (values may be seen in right triangle) (A1)

$(arctan (\frac{1}{\sqrt{3}}) =) \frac{π}{6}$ (seen anywhere) (accept degrees) (A1)

$2 x = \frac{π}{6}, \frac{7 π}{6}$

$x = \frac{π}{12}, \frac{7 π}{12}$ A1A1

Note: Do not award the final A1 if any additional solutions are seen.
Award A1A0 for correct answers in degrees.
Award A0A0 for correct answers in degrees with additional values.

[5 marks]

Examiners report

Determining the composite function was very well done. In part (b) very few candidates showed any recognition that tan (or cot) were required to solve this trigonometric equation. Many saw the 2x and simply employed one of the double angle rules but could not then progress to an answer.

[N/A]

Syllabus sections

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

Show 49 related questions

22M.1.SL.TZ2.1c:
Find the value of $x$ such that $f (x) = 0$ .
22M.2.SL.TZ1.8b.i:
Find $f^{- 1} (x)$ .
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.SL.TZ0.5b:
Find $f (4)$ .
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.
18N.3.AHL.TZ0.Hsrg_4b:
State with a reason whether or not $f$ and $g$ commute.
21N.1.SL.TZ0.5a:
Write down the value of $f' (4)$ .
21N.1.AHL.TZ0.2b.ii:
the $y$ -axis.
17N.1.SL.TZ0.S_5a:
Find $(g \circ f)(x)$ .
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.
19N.2.SL.TZ0.S_3b:
Let ${\text{C}}$ be a point on the graph of $h$ . The tangent to the graph of $h$ at ${\text{C}}$ is parallel to the graph of $f$ .

Find the $x$ -coordinate of ${\text{C}}$ .
19N.2.SL.TZ0.S_3a:
Find $h\left( x \right)$ .
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
18N.3.AHL.TZ0.Hsrg_4c:
Find the inverse of $f$ .
22M.1.SL.TZ2.1b:
Find $(f \circ g) (0)$ .
EXN.1.SL.TZ0.8c:
Find an expression for $f^{- 1} (x)$ , stating its domain.
22M.1.AHL.TZ2.11c:
Given that $(h \circ g) (a) = \frac{π}{4}$ , find the value of $a$ .

Give your answer in the form $p + \frac{q}{2} \sqrt{r}$ , where $p, q, r \in ℤ^{+}$ .
22M.2.SL.TZ1.8b.ii:
Using an algebraic approach, show that the graph of $f^{- 1}$ is obtained by a reflection of the graph of $f$ in the $y$ -axis followed by a reflection in the $x$ -axis.
20N.2.SL.TZ0.S_4b.i:
Solve the equation $(f \circ g) (x) = x$ .
21M.1.SL.TZ1.1a.i:
Write down the value of $f (2)$ .
21M.1.SL.TZ1.1a.ii:
Write down the value of $(f \circ f) (2)$ .
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
18N.3.AHL.TZ0.Hsrg_4a.i:
Find $\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .
SPM.1.SL.TZ0.5b:
Given that ${\left( {g \circ f} \right)^{ - 1}}\left( a \right) = 4$ , find the value of $a$ .
18N.3.AHL.TZ0.Hsrg_4a.ii:
Find $\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)$ .
17M.2.SL.TZ2.S_6a:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
20N.2.SL.TZ0.S_4b.ii:
Hence or otherwise, given that $g (2 a) = f^{- 1} (2 a)$ , find the value of $a$ .
EXN.1.SL.TZ0.5:
The functions $f$ and $g$ are defined for $x \in ℝ$ by $f (x) = x - 2$ and $g (x) = a x + b$ , where $a, b \in ℝ$ .

Given that $(f \circ g) (2) = - 3$ and $(g \circ f) (1) = 5$ , find the value of $a$ and the value of $b$ .
21M.2.AHL.TZ2.12d:
Find an expression for $g^{- 1} (x)$ , justifying your answer.
EXN.1.SL.TZ0.8b:
State the range of $f$ .
EXN.2.SL.TZ0.2c:
Arc $APB$ is twice the length of chord $[AB]$ .

Find the value of $θ$ .
21M.2.SL.TZ2.5b:
Given that $(g \circ f) (x) \leq 0$ for all $x \in ℝ$ , determine the set of possible values for $c$ .
SPM.1.SL.TZ0.5a:
Show that $\left( {g \circ f} \right)\left( x \right) = 2x + 11$ .
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 $a$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
21N.1.AHL.TZ0.2b.i:
the $x$ -axis.
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
21N.1.SL.TZ0.5c:
Find $h (4)$ .
16N.3.AHL.TZ0.Hsrg_2a:
(i) Sketch the graph of $y = f(x)$ and hence justify whether or not $f$ is a bijection.

(ii) Show that $A$ is a group under the binary operation of multiplication.

(iii) Give a reason why $B$ is not a group under the binary operation of multiplication.

(iv) Find an example to show that $f(a \times b) = f(a) \times f(b)$ is not satisfied for all $a,{\text{ }}b \in A$ .
20N.2.SL.TZ0.S_4a:
Find $(f \circ g) (x)$ .
16N.3.AHL.TZ0.Hsrg_2b:
(i) Sketch the graph of $y = g(x)$ and hence justify whether or not $g$ is a bijection.

(ii) Show that $g(a + b) = g(a) \times g(b)$ for all $a,{\text{ }}b \in \mathbb{R}$ .

(iii) Given that $\{ \mathbb{R},{\text{ }} + \}$ and $\{ D,{\text{ }} \times \}$ are both groups, explain whether or not they are isomorphic.
EXN.1.SL.TZ0.8d:
Find the coordinates of the point(s) where the graphs of $y = f (x)$ and $y = f^{- 1} (x)$ intersect.
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
21N.1.SL.TZ0.5d:
Hence find the equation of the tangent to the graph of $h$ at $x = 4$ .
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
21N.1.SL.TZ0.8d.ii:
Find the value of $p$ and the value of $q$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Hide related questions

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

View options