IBDP-2022-Functions-Dom
Ran [167 marks]
The following diagram shows the graph of y = f (x), −3 ⩽ x ⩽ 5.
1a. Find the value of (f ∘ f) (1). [2 marks]
1b. Given that f −1 (a) = 3, determine the value of a. [2 marks]
1c. Given that g (x) = 2f (x − 1), find the domain and range of g. [2 marks]
Consider the functions f and g defined by f (x) = ln |x| , x ∈ R \ {0}, and
g (x) = ln |x + k|, x ∈ R \ {−k}, where k ∈ R, k > 2.
2a. Describe the transformation by which f (x) is transformed to g (x). [1 mark]
2b. State the range of g . [1 mark]
2c. Sketch the graphs of y = f (x) and y = g (x) on the same axes, clearly [6 marks]
stating the points of intersection with any axes.
The graphs of f and g intersect at the point P .
2d. Find the coordinates of P. [2 marks]
y = f (x) at P passes through the origin (0, 0).
2e. The tangent to [7 marks]
Determine the value of k.
3a. Sketch the graph of y x−4 [5 marks]
= 2x−5
, stating the equations of any asymptotes
and the coordinates of any points of intersection with the axes.
Consider the function f : x → √ 2xx−4
−5
.
3b. Write down the largest possible domain of f . [2 marks]
3c. Write down the corresponding range of f . [1 mark]
The function f is defined by f (x) = sec x + 2, 0 ⩽ x < π
2.
4a. Write down the range of f . [1 mark]
4b. Find f −1 (x), stating its domain. [4 marks]
ax+b
The function f is defined by f (x) = cx+d
, for x ∈ R, x ≠ − dc .
5a. Find the inverse function f −1 , stating its domain. [5 marks]
2x−3
The function g is defined by g (x) = x−2
, x ∈ R, x ≠ 2
5b. Express g (x) in the form A + B
where A, B are constants. [2 marks]
x−2
5c. Sketch the graph of y = g (x). State the equations of any asymptotes [3 marks]
and the coordinates of any intercepts with the axes.
5d. The function h is defined by h (x) = √x, for x ≥ 0. [4 marks]
State the domain and range of h ∘ g .
3x2+10
The function f is given by f(x) = x2−4
,x ∈ R, x ≠ 2, x ≠ −2.
6a. Prove that f is an even function. [2 marks]
6b. Sketch the graph y = f(x). [3 marks]
6c. Write down the range of f . [2 marks]
= ( ) −3 ≤ ≤5
The following graph represents a function y = f(x), where −3 ≤ x ≤ 5.
The function has a maximum at (3, 1) and a minimum at (−1, − 1).
7a. The functions u and v are defined as u(x) = x − 3, v(x) = 2x where [7 marks]
x ∈ R.
(i) State the range of the function u ∘ f .
(ii) State the range of the function u ∘ v ∘ f .
(iii) Find the largest possible domain of the function f ∘ v ∘ u.
7b. (i) Explain whyf does not have an inverse. [6 marks]
(ii) The domain of f is restricted to define a function g so that it has an inverse
g −1 .
State the largest possible domain of g .
(iii) Sketch a graph of y = g −1 (x), showing clearly the y-intercept and stating
the coordinates of the endpoints.
2x−5 [8 marks]
7c. Consider the function defined by h(x) = x+ d
, x ≠ −d and d ∈ R.
(i) Find an expression for the inverse function h −1 (x).
(ii) Find the value of d such that h is a self-inverse function.
2x
For this value of d, there is a function k such that h ∘ k(x) = x+1
, x ≠ −1.
(iii) Find k(x).
3x−2
A function f is defined by f(x) = 2x−1
,x ∈ R, x ≠ 12 .
8a. Find an expression for f −1 (x). [4 marks]
8b. Given that f(x) can be written in the form f(x) = A + B
, find the [2 marks]
2x−1
values of the constants A and B.
8c. Hence, write down 3x−2 [1 mark]
∫ 2x−1
dx.
x+1
Consider the functions f(x) = tan x, 0 ≤ x < π
2 and g(x) = x−1
,x ∈ R, x ≠ 1.
9a. Find an expression for g ∘ f(x), stating its domain. [2 marks]
9b. Hence show that g ∘ f(x) sin x+cos x [2 marks]
= sin x−cos x
.
9c. Lety = g ∘ f(x), find an exact value for dy
dx
at the point on the graph of [6 marks]
y = g ∘ f(x) where x = π6 , expressing your answer in the form
a + b√3, a, b ∈ Z.
9d. Show that the area bounded by the graph of y = g ∘ f(x), the x-axis and [6 marks]
π
the lines x = 0 and x = 6
is ln(1 + √3).
10. A function f is defined by f(x) = x3 + ex + 1, x ∈ R. By considering [4 marks]
f ′ (x) determine whether f is a one-to-one or a many-to-one function.
Consider the following functions:
2x2 + 3
f(x) = ,x ⩾ 0
75
|3x − 4|
g(x) = ,x ∈ R .
10
11a. State the range of f and of g . [2 marks]
11b. Find an expression for the composite function f ∘ g(x) in the form [4 marks]
2+
bx+c
ax
3750
, where a, b and c ∈ Z .
−1
11c. (i) Find an expression for the inverse function f −1 (x) . [4 marks]
(ii) State the domain and range of f −1 .
11d. The domains of f and g are now restricted to {0, 1, 2, 3, 4} . [6 marks]
By considering the values of f and g on this new domain, determine which of f and
g could be used to find a probability distribution for a discrete random variable X ,
stating your reasons clearly.
11e. Using this probability distribution, calculate the mean of X . [2 marks]
Consider the equation yx2 + (y − 1)x + (y − 1) = 0.
12a. Find the set of values of y for which this equation has real roots. [4 marks]
12b. Hence determine the range of the function f x+1 [3 marks]
:x→ x2+x+1
.
12c. Explain why f has no inverse. [1 mark]
Consider the functions given below.
f(x) = 2x + 3
1
g(x) = ,x ≠ 0
x
13a. (i) Find (g ∘ f) (x) and write down the domain of the function. [2 marks]
(ii) Find (f ∘ g) (x) and write down the domain of the function.
13b. Find the coordinates of the point where the graph of y = f(x) and the [4 marks]
graph of y = (g −1 ∘ f ∘ g) (x) intersect.
( )= >0 ⩾0
14. A function is defined as f(x) = k√x, with k > 0 and x ⩾ 0 . [16 marks]
(a) Sketch the graph of y = f(x) .
(b) Show that f is a one-to-one function.
(c) Find the inverse function, f −1 (x) and state its domain.
(d) If the graphs of y = f(x) and y = f −1 (x) intersect at the point (4, 4) find
the value of k .
(e) Consider the graphs of y = f(x) and y = f −1 (x) using the value of k found
in part (d).
(i) Find the area enclosed by the two graphs.
(ii) The line x = c cuts the graphs of y = f(x) and y = f −1 (x) at the points P
and Q respectively. Given that the tangent to y = f(x) at point P is parallel to the
tangent to y = f −1 (x) at point Q find the value of c .
15. The functions f and g are defined as: [8 marks]
2
f(x) = ex , x ⩾ 0
1
g(x) = , x ≠ −3.
x+3
(a) Find h(x) where h(x) = g ∘ f(x) .
(b) State the domain of h −1 (x) .
(c) Find h −1 (x) .
16. Let 4 [6 marks]
f(x) = x+2
, x ≠ −2 and g(x) = x − 1.
If h = g ∘ f , find
(a) h(x) ;
(b) h−1 (x) , where h−1 is the inverse of h.
© International Baccalaureate Organization 2020
International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®
Printed for The Doon School, Dehradun (2603)
