Functions- recap May 2025 [139 marks]

1. [Maximum mark: 5] 24M.1.SL.TZ2.3

(a) Solve 3m2 + 5m − 2 = 0. [3]

(b) Hence or otherwise, solve 3 × 9x + 5 × 3x − 2 = 0. [2]

2. [Maximum mark: 8] 24M.1.SL.TZ2.6

2(x+3)
A function f is defined by f (x) =
3(x+2)
, where x ∈ R, x ≠ −2.

The graph y = f (x) is shown below.

(a) Write down the equation of the horizontal asymptote. [1]

 Consider g(x) = mx + 1, where m ∈ R, m ≠ 0.

(b.i) Write down the number of solutions to f (x) = g(x) for

m > 0. [1]

(b.ii) Determine the value of m such that f (x) = g(x) has only

one solution for x. [4]

(b.iii) Determine the range of values for m, where f (x) = g(x)

has two solutions for x ≥ 0. [2]

3. [Maximum mark: 17] 24M.1.SL.TZ2.8

2x+2
The derivative of a function f is given by f ′(x) = 2
x +2x+2
, for x ∈ R.

(a.i) Show that x2 + 2x + 2 > 0 for all values of x. [2]

(a.ii) Hence, find the values of x for which f is increasing. [1]

(b.i) Write down the value of x for which f ′(x) = 0. [1]

(b.ii) Show that f ′′(x) =

2
−2x −4x
.
2
(x +2x+2)
2
[4]

(b.iii) Hence, justify that the value of x found in part (b)(i)

corresponds to a local minimum point on the graph of f . [2]

It is given that f (2) = 3 + ln 10.

(c) Find an expression for f (x). [4]

(d) Find the equation of the normal to the graph of f at

(2, 3 + ln 10). [3]
4. [Maximum mark: 12] 24M.2.SL.TZ1.7
Consider the function defined by f (x) =
3

2
x−2
e , 0 ≤ x ≤ 4.

(a) Show that the inverse function is given by

2x
). [3]
−1
f (x) = 2 + ln (
3

The graphs of f and f −1 intersect at two points P and Q, as shown on the

following diagram.

(b) Find PQ. [3]

The graph of f is reflected in the x-axis and then translated parallel to the y-axis
by 5 units in the positive direction to give the graph of a function g.

(c) Write down

(c.i) an expression for g(x); [2]

(c.ii) the domain of g. [1]

 (d) Solve the equation f (x) = g(x). Give your answer in the
form x = a + ln b, where a, b ∈ Q. [3]

5. [Maximum mark: 6] 24M.2.SL.TZ2.4

The loudness of a sound, L, measured in decibels, is related to its intensity, I
units, by L = 10 log10 (I × 10
12
).

Consider two sounds, S1 and S2.

S1 has an intensity of 10−6 units and a loudness of 60 decibels.

S2 has an intensity that is twice that of S1.

(a) State the intensity of S2. [1]

(b) Determine the loudness of S2. [2]

The maximum loudness of thunder in a thunderstorm was measured to be 115

decibels.

(c) Find the corresponding intensity, I , of the thunder. [3]

6. [Maximum mark: 5] 23N.1.SL.TZ1.2

Consider the functions f (x) = x + 2 and g(x) = x
2
− k , where k is a
2

real constant.

(a) Write down an expression for (g ∘ f ) (x). [2]

(b) Given that (g ∘ f )(4) = 11 , find the possible values of k. [3]

7. [Maximum mark: 15] 23N.1.SL.TZ1.8

The functions f and g are defined by

f (x) = ln (2x − 7), where x >

7

g(x) = 2 ln x − ln d, where x > 0, d ∈ R .

+

(a) State the equation of the vertical asymptote to the graph of

y = g(x). [1]

The graphs of y = f (x) and y = g(x) intersect at two distinct points.

(b.i) Show that, at the points of intersection, x2 − 2dx + 7d = 0

. [4]

(b.ii) Hence, show that d2 − 7d > 0. [3]

(b.iii) Find the range of possible values of d. [2]

The following diagram shows parts of the graph y = f (x) and y = g(x).

The graphs intersect at x = p and x = q, where p < q.

 (c) In the case where d = 10, find the value of q − p. Express

your answer in the form a√b, where a, b ∈ Z

+
. [5]

8. [Maximum mark: 7] 23N.2.SL.TZ1.1

Consider the function defined by f (x) = x
2
− 10x. The graph of f passes
through the point A(4, −24).

(a.i) Find the gradient of the tangent to the graph of f at the point
A. [2]

(a.ii) Hence, write down the gradient of the normal to the graph of f
at Point A. [1]

(b) Write down the equation of the normal to the graph of f at

Point A. [1]

The normal to the graph of f at point A intersects the graph of f again at a

second point B.

(c) Find the coordinates of B. [3]

9. [Maximum mark: 5] 23N.2.SL.TZ1.3

Consider the function f (x) = e
x
− 2x − 5.

(a) On the following axes, sketch the graph of f for

−4 ≤ x ≤ 3.
 [3]

The function g is defined by g(x) = e

3x
− 6x − 7.

(b) The graph of g is obtained from the graph of f by a horizontal

stretch with scale factor k , followed by a vertical translation of
c units.

Find the value of k and the value of c. [2]

10. [Maximum mark: 6] 23M.1.SL.TZ1.5

Find the range of possible values of k such that e 2x
+ ln k = 3e
x

has at least one real solution. [6]

11. [Maximum mark: 7] 23M.1.SL.TZ1.2

The function f is defined by f (x) for x ∈ R, x ≠ 2.
7x+7
=
2x−4

(a) Find the zero of f (x). [2]

 (b) For the graph of y = f (x), write down the equation of

(b.i) the vertical asymptote; [1]

(b.ii) the horizontal asymptote. [1]

(c) Find f −1(x), the inverse function of f (x). [3]

12. [Maximum mark: 13] 23M.1.SL.TZ1.7

The function h is defined by h(x) = 2xe + 3, for x ∈ R. The following
x

diagram shows part of the graph of h, which has a local minimum at point A.

(a) Find the value of the y-intercept. [2]

(b) Find h′(x). [2]

(c) Hence, find the coordinates of A. [5]

(d.i) Show that h′′(x) = (2x + 4)e .

x
[2]
 (d.ii) Find the values of x for which the graph of h is concave-up. [2]

13. [Maximum mark: 5] 23M.1.SL.TZ1.1

Point P has coordinates (−3, 2), and point Q has coordinates (15, − 8).

Point M is the midpoint of [PQ] .

(a) Find the coordinates of M. [2]

Line L is perpendicular to [PQ] and passes through M.

(b) Find the gradient of L. [2]

(c) Hence, write down the equation of L. [1]

14. [Maximum mark: 7] 23M.1.SL.TZ2.6

The functions f and g are defined for x ∈ R by

f (x) = ax + b, where a, b ∈ Z

+ x + 3.
2
g(x) = x

Find the two possible functions f such that

− 14x + 15.
2
(g ∘ f )(x) = 4x

[7]

15. [Maximum mark: 5] 23M.1.SL.TZ2.3

1
A function f is defined by f (x) = 1 −
x−2
, where x ∈ R, x ≠ 2.
 (a) The graph of y = f (x) has a vertical asymptote and a
horizontal asymptote.

Write down the equation of

(a.i) the vertical asymptote; [1]

(a.ii) the horizontal asymptote. [1]

(b) Find the coordinates of the point where the graph of

y = f (x) intersects

(b.i) the y-axis; [1]

(b.ii) the x-axis. [1]

(c) On the following set of axes, sketch the graph of y = f (x),

showing all the features found in parts (a) and (b).

[1]

16. [Maximum mark: 16] 23M.1.SL.TZ2.7

The following diagram shows part of the graph of a quadratic function f .

The vertex of the parabola is (−2, − 5) and the y-intercept is at point P.

(a) Write down the equation of the axis of symmetry. [1]

1 2
The function can be written in the form f (x) =
4
(x − h) + k, where h,

k ∈ Z.

(b) Write down the values of h and k. [2]

(c) Find the y-coordinate of P. [2]

In the following diagram, the line L is normal to the graph of f at point P.

(d) Find the equation of the line L, in the form y = ax + b. [3]

The line L intersects the graph of f at a second point, Q, as shown above.

(e) Calculate the distance between P and Q. [8]

© International Baccalaureate Organization, 2025