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Browse Functions
37 questions at your level
Difficulty
All
Level 2
Level 3
Level 4
Level 5
Function notation
14 questions
Lesson
Not started
Mark as done
f
(
x
)
=
4
x
+
1
f(x) = 4x + 1
f
(
x
)
=
4
x
+
1
(a)
Work out
f
(
3
)
f(3)
f
(
3
)
(b)
Work out
f
(
−
1
)
f(-1)
f
(
−
1
)
(c)
Solve
f
(
x
)
=
29
f(x) = 29
f
(
x
)
=
29
●●
●●●
Level 2
4 marks
Start
→
Mark as done
f
(
x
)
=
3
x
−
4
f(x) = 3x - 4
f
(
x
)
=
3
x
−
4
(a)
Work out
f
(
5
)
f(5)
f
(
5
)
(b)
Work out
f
(
−
2
)
f(-2)
f
(
−
2
)
(c)
Solve
f
(
x
)
=
20
f(x) = 20
f
(
x
)
=
20
●●●
●●
Level 3
4 marks
Start
→
Mark as done
h
(
x
)
=
x
−
3
2
h(x) = \dfrac{x - 3}{2}
h
(
x
)
=
2
x
−
3
(a)
Work out
h
(
11
)
h(11)
h
(
11
)
(b)
Solve
h
(
x
)
=
−
5
h(x) = -5
h
(
x
)
=
−
5
●●●
●●
Level 3
3 marks
Start
→
Mark as done
g
(
x
)
=
10
−
3
x
g(x) = 10 - 3x
g
(
x
)
=
10
−
3
x
(a)
Work out
g
(
4
)
g(4)
g
(
4
)
(b)
Solve
g
(
x
)
=
−
8
g(x) = -8
g
(
x
)
=
−
8
●●●
●●
Level 3
3 marks
Start
→
Mark as done
f
(
x
)
=
5
x
−
4
\mathrm{f}(x) = 5x - 4
f
(
x
)
=
5
x
−
4
(a)
Work out
f
(
3
)
\mathrm{f}(3)
f
(
3
)
. [1]
(b)
Find
f
−
1
(
x
)
\mathrm{f}^{-1}(x)
f
−
1
(
x
)
. [2]
(c)
Solve
f
(
x
)
=
f
−
1
(
x
)
\mathrm{f}(x) = \mathrm{f}^{-1}(x)
f
(
x
)
=
f
−
1
(
x
)
. [1]
●●●
●●
Level 3
4 marks
Start
→
Mark as done
g
(
x
)
=
x
2
+
1
g(x) = x^2 + 1
g
(
x
)
=
x
2
+
1
(a)
Work out
g
(
−
3
)
g(-3)
g
(
−
3
)
(b)
Solve
g
(
x
)
=
50
g(x) = 50
g
(
x
)
=
50
●●●●
●
Level 4
3 marks
Start
→
Mark as done
f
(
x
)
=
5
−
2
x
f(x) = 5 - 2x
f
(
x
)
=
5
−
2
x
(a)
Work out
f
(
0.5
)
f(0.5)
f
(
0.5
)
(b)
Solve
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
●●●●
●
Level 4
3 marks
Start
→
Mark as done
f
(
x
)
=
x
2
−
3
x
f(x) = x^2 - 3x
f
(
x
)
=
x
2
−
3
x
(a)
Work out
f
(
5
)
f(5)
f
(
5
)
(b)
Solve
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
●●●●
●
Level 4
3 marks
Start
→
Mark as done
h
(
x
)
=
12
x
h(x) = \dfrac{12}{x}
h
(
x
)
=
x
12
for
x
≠
0
x \neq 0
x
=
0
(a)
Work out
h
(
8
)
h(8)
h
(
8
)
(b)
Work out
h
(
−
4
)
h(-4)
h
(
−
4
)
(c)
Solve
h
(
x
)
=
5
h(x) = 5
h
(
x
)
=
5
●●●●
●
Level 4
4 marks
Start
→
Mark as done
f
(
x
)
=
x
2
+
2
x
f(x) = x^2 + 2x
f
(
x
)
=
x
2
+
2
x
(a)
Work out
f
(
−
3
)
f(-3)
f
(
−
3
)
(b)
Solve
f
(
x
)
=
15
f(x) = 15
f
(
x
)
=
15
●●●●
●
Level 4
5 marks
Start
→
Mark as done
The functions f and g are given by
f
(
x
)
=
18
x
−
2
and
g
(
x
)
=
7
−
4
x
\mathrm{f}(x) = \frac{18}{x - 2} \qquad \text{and} \qquad \mathrm{g}(x) = 7 - 4x
f
(
x
)
=
x
−
2
18
and
g
(
x
)
=
7
−
4
x
(a)
Find
f
(
−
4
)
\mathrm{f}(-4)
f
(
−
4
)
(1 mark)
(b)
Find
f
g
(
−
1
)
\mathrm{fg}(-1)
fg
(
−
1
)
(2 marks)
(c)
Find
g
−
1
(
15
)
\mathrm{g}^{-1}(15)
g
−
1
(
15
)
(2 marks)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
f
(
x
)
=
2
x
−
3
f(x) = 2x - 3
f
(
x
)
=
2
x
−
3
and
g
(
x
)
=
x
+
4
g(x) = x + 4
g
(
x
)
=
x
+
4
(a)
Solve
f
(
x
)
=
g
(
x
)
f(x) = g(x)
f
(
x
)
=
g
(
x
)
(b)
The graphs of
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
and
y
=
g
(
x
)
y = g(x)
y
=
g
(
x
)
are drawn on the same axes. What does your answer to part (a) tell you about the two graphs?
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
3
g
(
x
)
=
4
x
2
−
5
\mathrm{f}(x) = 2x + 3 \qquad \mathrm{g}(x) = 4x^2 - 5
f
(
x
)
=
2
x
+
3
g
(
x
)
=
4
x
2
−
5
Show that
g
f
(
x
)
−
2
f
g
(
x
)
=
0
\mathrm{gf}(x) - 2\mathrm{fg}(x) = 0
gf
(
x
)
−
2
fg
(
x
)
=
0
has just one solution for
x
x
x
.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
f
(
x
)
=
5
x
−
4
h
(
x
)
=
2
x
2
+
x
\mathrm{f}(x) = 5x - 4 \qquad \mathrm{h}(x) = 2x^2 + x
f
(
x
)
=
5
x
−
4
h
(
x
)
=
2
x
2
+
x
Show that
5
f
h
(
x
)
−
h
f
(
x
)
=
0
5\mathrm{fh}(x) - \mathrm{hf}(x) = 0
5
fh
(
x
)
−
hf
(
x
)
=
0
has just one solution for
x
x
x
.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Composite functions
16 questions
Lesson
Not started
Mark as done
f
(
x
)
=
x
−
1
f(x) = x - 1
f
(
x
)
=
x
−
1
and
g
(
x
)
=
4
x
g(x) = 4x
g
(
x
)
=
4
x
(a)
Work out
g
f
(
5
)
gf(5)
g
f
(
5
)
(b)
Work out
f
g
(
5
)
fg(5)
f
g
(
5
)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
f
(
x
)
=
3
x
−
2
f(x) = 3x - 2
f
(
x
)
=
3
x
−
2
and
g
(
x
)
=
x
+
5
g(x) = x + 5
g
(
x
)
=
x
+
5
(a)
Work out
f
g
(
2
)
fg(2)
f
g
(
2
)
(b)
Work out
g
f
(
2
)
gf(2)
g
f
(
2
)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
The functions
f
\mathrm{f}
f
and
g
\mathrm{g}
g
are such that
f
(
x
)
=
3
x
−
2
g
(
x
)
=
x
2
+
1
\mathrm{f}(x) = 3x - 2 \qquad \mathrm{g}(x) = x^2 + 1
f
(
x
)
=
3
x
−
2
g
(
x
)
=
x
2
+
1
(a)
Find
f
g
(
4
)
\mathrm{fg}(4)
fg
(
4
)
(2 marks)
(b)
Find
f
−
1
(
x
)
\mathrm{f}^{-1}(x)
f
−
1
(
x
)
(2 marks)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
1
f(x) = 2x + 1
f
(
x
)
=
2
x
+
1
and
g
(
x
)
=
x
2
−
3
g(x) = x^2 - 3
g
(
x
)
=
x
2
−
3
(a)
Work out
f
g
(
3
)
fg(3)
f
g
(
3
)
(b)
Work out
g
f
(
3
)
gf(3)
g
f
(
3
)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
f
(
x
)
=
x
2
f(x) = x^2
f
(
x
)
=
x
2
and
g
(
x
)
=
2
x
−
1
g(x) = 2x - 1
g
(
x
)
=
2
x
−
1
(a)
Find
f
g
(
x
)
fg(x)
f
g
(
x
)
(b)
Show that
f
g
(
2
)
≠
g
f
(
2
)
fg(2) \neq gf(2)
f
g
(
2
)
=
g
f
(
2
)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
f
(
x
)
=
12
x
f(x) = \dfrac{12}{x}
f
(
x
)
=
x
12
for
x
≠
0
x \neq 0
x
=
0
, and
g
(
x
)
=
x
+
2
g(x) = x + 2
g
(
x
)
=
x
+
2
(a)
Work out
f
g
(
4
)
fg(4)
f
g
(
4
)
(b)
Work out
g
f
(
4
)
gf(4)
g
f
(
4
)
(c)
Solve
f
g
(
x
)
=
3
fg(x) = 3
f
g
(
x
)
=
3
●●●●
●
Level 4
6 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
3
g
(
x
)
=
x
2
−
1
\mathrm{f}(x) = 2x + 3 \qquad \mathrm{g}(x) = x^2 - 1
f
(
x
)
=
2
x
+
3
g
(
x
)
=
x
2
−
1
(a)
Show that
g
f
(
x
)
=
4
x
2
+
12
x
+
8
\mathrm{gf}(x) = 4x^2 + 12x + 8
gf
(
x
)
=
4
x
2
+
12
x
+
8
(2 marks)
(b)
Solve
g
f
(
x
)
=
8
\mathrm{gf}(x) = 8
gf
(
x
)
=
8
(2 marks)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
f
(
x
)
=
3
x
−
1
g
(
x
)
=
x
2
+
1
\mathrm{f}(x) = 3x - 1 \qquad \mathrm{g}(x) = x^2 + 1
f
(
x
)
=
3
x
−
1
g
(
x
)
=
x
2
+
1
(a)
Work out
f
g
(
2
)
\mathrm{fg}(2)
fg
(
2
)
. [2]
(b)
Solve
f
g
(
x
)
=
g
f
(
x
)
\mathrm{fg}(x) = \mathrm{gf}(x)
fg
(
x
)
=
gf
(
x
)
. [3]
●●●●
●
Level 4
5 marks
Start
→
Mark as done
The functions f and g are given by
f
(
x
)
=
18
x
−
2
and
g
(
x
)
=
7
−
4
x
\mathrm{f}(x) = \frac{18}{x - 2} \qquad \text{and} \qquad \mathrm{g}(x) = 7 - 4x
f
(
x
)
=
x
−
2
18
and
g
(
x
)
=
7
−
4
x
(a)
Find
f
(
−
4
)
\mathrm{f}(-4)
f
(
−
4
)
(1 mark)
(b)
Find
f
g
(
−
1
)
\mathrm{fg}(-1)
fg
(
−
1
)
(2 marks)
(c)
Find
g
−
1
(
15
)
\mathrm{g}^{-1}(15)
g
−
1
(
15
)
(2 marks)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
1
f(x) = 2x + 1
f
(
x
)
=
2
x
+
1
and
g
(
x
)
=
x
2
−
3
g(x) = x^2 - 3
g
(
x
)
=
x
2
−
3
(a)
Find
f
g
(
x
)
fg(x)
f
g
(
x
)
, giving your answer in its simplest form.
(b)
Find
g
f
(
x
)
gf(x)
g
f
(
x
)
, giving your answer in its simplest form.
●●●●●
Level 5
4 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
1
f(x) = 2x + 1
f
(
x
)
=
2
x
+
1
and
g
(
x
)
=
x
2
−
3
g(x) = x^2 - 3
g
(
x
)
=
x
2
−
3
Solve
f
g
(
x
)
=
45
fg(x) = 45
f
g
(
x
)
=
45
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
1
f(x) = 2x + 1
f
(
x
)
=
2
x
+
1
(a)
Find
f
f
(
x
)
ff(x)
f
f
(
x
)
, giving your answer in its simplest form.
(b)
Solve
f
f
(
x
)
=
23
ff(x) = 23
f
f
(
x
)
=
23
●●●●●
Level 5
4 marks
Start
→
Mark as done
f
(
x
)
=
x
+
3
2
f(x) = \dfrac{x + 3}{2}
f
(
x
)
=
2
x
+
3
and
g
(
x
)
=
4
x
−
1
g(x) = 4x - 1
g
(
x
)
=
4
x
−
1
(a)
Find
f
g
(
x
)
fg(x)
f
g
(
x
)
, giving your answer in its simplest form.
(b)
Solve
f
g
(
x
)
=
17
fg(x) = 17
f
g
(
x
)
=
17
●●●●●
Level 5
4 marks
Start
→
Mark as done
f
(
x
)
=
5
−
x
f(x) = 5 - x
f
(
x
)
=
5
−
x
(a)
Find
f
f
(
x
)
ff(x)
f
f
(
x
)
, giving your answer in its simplest form.
(b)
What does your answer to part (a) tell you about the function
f
f
f
?
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
3
g
(
x
)
=
4
x
2
−
5
\mathrm{f}(x) = 2x + 3 \qquad \mathrm{g}(x) = 4x^2 - 5
f
(
x
)
=
2
x
+
3
g
(
x
)
=
4
x
2
−
5
Show that
g
f
(
x
)
−
2
f
g
(
x
)
=
0
\mathrm{gf}(x) - 2\mathrm{fg}(x) = 0
gf
(
x
)
−
2
fg
(
x
)
=
0
has just one solution for
x
x
x
.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Mark as done
f
(
x
)
=
5
x
−
4
h
(
x
)
=
2
x
2
+
x
\mathrm{f}(x) = 5x - 4 \qquad \mathrm{h}(x) = 2x^2 + x
f
(
x
)
=
5
x
−
4
h
(
x
)
=
2
x
2
+
x
Show that
5
f
h
(
x
)
−
h
f
(
x
)
=
0
5\mathrm{fh}(x) - \mathrm{hf}(x) = 0
5
fh
(
x
)
−
hf
(
x
)
=
0
has just one solution for
x
x
x
.
(5 marks)
●●●●●
Level 5
5 marks
Start
→
Inverse functions
13 questions
Lesson
Not started
Mark as done
The functions
f
\mathrm{f}
f
and
g
\mathrm{g}
g
are such that
f
(
x
)
=
3
x
−
2
g
(
x
)
=
x
2
+
1
\mathrm{f}(x) = 3x - 2 \qquad \mathrm{g}(x) = x^2 + 1
f
(
x
)
=
3
x
−
2
g
(
x
)
=
x
2
+
1
(a)
Find
f
g
(
4
)
\mathrm{fg}(4)
fg
(
4
)
(2 marks)
(b)
Find
f
−
1
(
x
)
\mathrm{f}^{-1}(x)
f
−
1
(
x
)
(2 marks)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
f
(
x
)
=
5
x
−
4
\mathrm{f}(x) = 5x - 4
f
(
x
)
=
5
x
−
4
(a)
Work out
f
(
3
)
\mathrm{f}(3)
f
(
3
)
. [1]
(b)
Find
f
−
1
(
x
)
\mathrm{f}^{-1}(x)
f
−
1
(
x
)
. [2]
(c)
Solve
f
(
x
)
=
f
−
1
(
x
)
\mathrm{f}(x) = \mathrm{f}^{-1}(x)
f
(
x
)
=
f
−
1
(
x
)
. [1]
●●●
●●
Level 3
4 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
5
f(x) = 2x + 5
f
(
x
)
=
2
x
+
5
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
●●●●
●
Level 4
2 marks
Start
→
Mark as done
g
(
x
)
=
x
−
4
3
g(x) = \dfrac{x - 4}{3}
g
(
x
)
=
3
x
−
4
(a)
Find
g
−
1
(
x
)
g^{-1}(x)
g
−
1
(
x
)
(b)
Work out
g
−
1
(
2
)
g^{-1}(2)
g
−
1
(
2
)
●●●●
●
Level 4
3 marks
Start
→
Mark as done
f
(
x
)
=
x
4
+
3
f(x) = \dfrac{x}{4} + 3
f
(
x
)
=
4
x
+
3
(a)
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
(b)
Work out
f
−
1
(
5
)
f^{-1}(5)
f
−
1
(
5
)
●●●●
●
Level 4
3 marks
Start
→
Mark as done
g
(
x
)
=
3
(
x
−
2
)
g(x) = 3(x - 2)
g
(
x
)
=
3
(
x
−
2
)
(a)
Find
g
−
1
(
x
)
g^{-1}(x)
g
−
1
(
x
)
(b)
Solve
g
−
1
(
x
)
=
10
g^{-1}(x) = 10
g
−
1
(
x
)
=
10
●●●●
●
Level 4
4 marks
Start
→
Mark as done
f
(
x
)
=
6
x
−
1
f(x) = 6x - 1
f
(
x
)
=
6
x
−
1
(a)
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
(b)
Show that
f
(
f
−
1
(
x
)
)
=
x
f\!\left(f^{-1}(x)\right) = x
f
(
f
−
1
(
x
)
)
=
x
●●●●
●
Level 4
4 marks
Start
→
Mark as done
The functions f and g are given by
f
(
x
)
=
18
x
−
2
and
g
(
x
)
=
7
−
4
x
\mathrm{f}(x) = \frac{18}{x - 2} \qquad \text{and} \qquad \mathrm{g}(x) = 7 - 4x
f
(
x
)
=
x
−
2
18
and
g
(
x
)
=
7
−
4
x
(a)
Find
f
(
−
4
)
\mathrm{f}(-4)
f
(
−
4
)
(1 mark)
(b)
Find
f
g
(
−
1
)
\mathrm{fg}(-1)
fg
(
−
1
)
(2 marks)
(c)
Find
g
−
1
(
15
)
\mathrm{g}^{-1}(15)
g
−
1
(
15
)
(2 marks)
●●●●
●
Level 4
5 marks
Start
→
Mark as done
h
(
x
)
=
5
x
−
2
h(x) = 5x - 2
h
(
x
)
=
5
x
−
2
Solve
h
−
1
(
x
)
=
3
h^{-1}(x) = 3
h
−
1
(
x
)
=
3
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
4
−
x
f(x) = 4 - x
f
(
x
)
=
4
−
x
(a)
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
(b)
What do you notice about
f
f
f
and
f
−
1
f^{-1}
f
−
1
?
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
3
x
−
1
x
+
2
f(x) = \dfrac{3x - 1}{x + 2}
f
(
x
)
=
x
+
2
3
x
−
1
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
10
x
−
3
f(x) = \dfrac{10}{x - 3}
f
(
x
)
=
x
−
3
10
for
x
≠
3
x \neq 3
x
=
3
Find
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
●●●●●
Level 5
3 marks
Start
→
Mark as done
f
(
x
)
=
2
x
+
c
f(x) = 2x + c
f
(
x
)
=
2
x
+
c
, where
c
c
c
is a constant.
f
−
1
(
11
)
=
4
f^{-1}(11) = 4
f
−
1
(
11
)
=
4
Work out the value of
c
c
c
.
●●●●●
Level 5
3 marks
Start
→