Gr 12 Maths – Functions
GR 12 MATHS
FUNCTIONS
QUESTIONS and ANSWERS
Work through the Grade 11 Functions downloads first to ensure
your foundation is solid before attempting inverse Functions.
We wish you the best of luck for your exams.
From
The Answer Series team
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1
Gr 12 Maths – Functions: Questions
2.
INVERSE FUNCTIONS
(Gr 12 only)
• NB: The INVERSE of a function REVERSES the process of
a function.
Determine the equation of the inverse of f in the form
1
-2
2.5
f (2) = 0
3.1
3.2
B.
1
O
x
2
O
5.4
D.
y
1
y
2
x
O
-2
5.5
(2)
1.2 Consider the graph of f given above.
1.2.1 Write down the equation of the given function and of
the inverse function in the form y = . . .
(2)
1.2.2 Hence complete: (a) f (x) = . . . (b)
f -1(x)
=...
(2)
-1
1.2.3 Show how the equation of f could have been
calculated from the equation of f.
1.2.4 Explain why f and f -1 are both one-to-one
relations.
Copyright © The Answer
Sketch the graphs of f and f -1 on the same set of axes,
indicating also the line y = x.
(4)
4.2
1
g ( x)
⎞
4.3 g ⎛⎜ 1
⎟
⎝ x⎠
x
Sketch the graph y = 2 , indicating the coordinates of
any three points on the graph.
Use the three points on the sketch to write down the
coordinates of three points on the inverse function of
x
y=2 .
On the same system of axes, sketch the inverse function
x
of y = 2 and the line y = x.
x
Describe the transformation from the graph y = 2 to its
inverse in words and give the rule for this transformation.
(2)
(4)
f -1 (x)
(b) Determine
in each case. Sketch the 2 cases.
Include the line y = x.
(4)
(c) Determine the domain and the range of f -1 in each
case.
Consider the function f where f (x) = - x
7.1
Write down the domain and range of f.
7.2
Determine the inverse function f -1 in the form f -1 (x) = . . . (2)
7.3
Sketch the graphs of the functions f , f -1 and the line
y = x on the same set of axes. What do you notice?
(3)
(3)
(3)
Write down the domain and the range of the graphs of :
x
y
(a) y = 2
(b) x = 2
6.1
Write down the domain and range of f.
Is f
8.1
Write down the coordinates of the reflections of the
following points in the line y = x :
(4)
an increasing or a decreasing function?
(1)
P(0; 0), Q(- 1;1), R(2; 4) and S(3; 9).
8.2
(4)
2
(2)
Determine the equation of g in the form y = . . .
(2)
2
(2)
8.3
Which of the points in Question 8.1 lie on the graph of f ? (2)
8.4
f and its inverse function, f -1, are reflections in the line. . .? (1)
8.5
Draw the graph f -1, the inverse function of f , on the
same system of axes.
(2)
8.6
Which of the points in Question 8.1 (question or answer)
lie on the graph of f -1 ?
(2)
Complete : (a) f (3) = . . .
(b) f -1 (9) = . . .
8.8
(c)
f (2) = 4
(d)
f -1 (4)
=2
(
) lies on f
;
(
;
) lies on f -1
Determine the equation of f -1 in the form f -1 (x) = . . .
f -1
8.9
Is
9.
If f (x) = (x + 2) ; x [ - 2, then f -1 (x) is equal to
a function? Give a reason for your answer.
A. x - 2
10.
(4)
(2)
(2)
2
2
(3)
(4)
Draw a sketch of the graph f which has equation y = x
for x m 0.
8.7
Sketch the graph of f and g on the same set of axes
where g is the reflection of f in the line y = x.
Draw the line y = x on the sketch.
2
(2)
7.4
(2)
5.7
Consider the function f where f (x) = 2x
-1
and x m 0.
Let the images be P ′, Q ′, R ′ and S ′ respectively.
(2)
6.
2
(4)
7.
(6)
Are both the above graphs functions? Why (not)?
6.3
(a) Name 2 ways in which the domain of f could be
restricted to ensure that the inverse is a function.
Sketch the 2 cases.
Write down the equation of f -1 in the form f -1 (x) = . . . . (2)
5.6
6.2
6.5
) lies on f -1 . (2)
(2)
(2)
(2)
;
Write down the equation of the inverse function in the
form x = . . .
[ In Topic 4, you will convert this equation to y = . . . ]
x
1
) lies on f and (
Given : g (x) = 3x - 2. Determine each of the following :
5.3
x
O
-1
;
4.
2
(2)
(1)
Are f and f -1 both functions? Why (not)?
5.2
y
(
= ...?
3.4
5.1
Is g the inverse function of f ? Why (not)?
(3)
Write down the coordinates of the x- and y-intercepts of the
4.1 g - 1 (x)
1.1 If the inverse of f is the reflection of f in the line y = x,
then the graph of the inverse is :
y
-1 (3)
function f (x) = 2x + 6 and of f -1, the inverse function of f. (4)
x
O
Sketch the graphs of the functions f, f -1 and y = x on
the same set of axes. What do you notice?
If f (1) = 3, then f
f
The graph of f is . . .
(2)
6.4
(2)
2.4
3.3
y
C.
Write down the domain and range of f.
2.2
2.3
• Concepts and techniques involving the general
characteristics of functions (first part of this topic) should be
thoroughly mastered before this section on inverse functions.
In particular, work through Questions 2 → 7.
A.
2.1
f -1 (x) = . . .
QUESTIONS
1.
Consider the function f (x) = - 3x + 6.
B. ±
x -2
C.
x-2
D. -
x -2
(2)
Given h - 1 (x) = - x . Then the equation of h is y = . . .
2
A. x ; x ≤ 0
2
B. x ; x ≥ 0
C. x
2
D.
x
(2)
Gr 12 Maths – Functions: Questions
11.
Given : g (x) = -1 +
x.
Determine the inverse of g (x) in the form g -1 (x) = . . .
12.
y
f (x) = - 2(x - 3)(x + 1) and
NOTES
(4)
D
h
h (x) = m x + c
S(2; a)
E
The graphs of f and h have
a common x-intercept at Q and
a common y-intercept at E.
f
The turning point of f is at D
and S(2; a) is a point on f.
O
x
Q
12.1 Calculate the coordinates of E and D.
(4)
12.2 Write down the coordinates of Q.
(1)
12.3 Determine the numerical values of :
(a) m
(b) a
(2)(2)
12.4 Write down the coordinates of the turning point of
f -1 , the inverse of f.
(2)
12.5 Is f -1 a function? Why (not)?
13.
(2)
The graph of the function g is shown below.
y
6-
4g
2-
-4
-2
O
2
4
6
x
-2 -
-4 13.1 Determine the domain and range of the function.
(2)
13.2 On this set of axes, draw the graph of the inverse
function of g.
(4)
13.3 Explain why this inverse is a function.
(1)
* Now do Paper E1 Q10 in Section 2 of this book.
* The Topic Guide indicates the examples in all the papers.
* See the end of Topic 4 for mixed questions (including
exponential and log functions) on inverse functions.
Copyright © The Answer
3
Gr 12 Maths – Functions: Answers
NB : The INVERSE of a function reverses the process of a function.
INVERSE FUNCTIONS
So : • f maps x = 1 onto y = 3
whereas
& 5.3
f -1 maps x = 3 onto y = 1
(Gr 12 only)
x
5.1
y=2
x
-3
0
3
1
8
1
8
y=2
y
f
O
5.2 (8 ; 3), (1 ; 0) and
f:
1.
y = 2x + 6
The given graph has points (- 2 ; 0) & (0 ; 1)
x-int. :
â The graph of the inverse has points (0 ; - 2) & (1 ; 0)
(put y = 0)
â 2x = - 6
â x = -3
â (- 3; 0) Given : y = 1 x + 1 C : y = 2x - 2 2
(b) f -1 (x) = 2x - 2 2
x = 1y+1
∴ Equation of f -1 :
% 2)
2
. . . swop x & y
â For f -1 (the inverse function)
x-int. : (6; 0) . . . make y the subject
y-int. :
Equation of f :
Range : y ∈ » 2
y = - 3x + 6
â
3
∴
3
f
3
y=x
f -1
2
O
2
6
x
We notice that f and f
are reflections in the
line y = x.
2.4 f (1) = 3 f -1 (3) = 1 Copyright © The Answer
6.1 Domain : x ∈ » Range : y m 0 ; y ∈ » ;
y
6.2
x-3 y=x
f
Equation of g :
y = 3x - 2
x = 3y - 2
â
4.3
and (0 ; 2) lies on f -1 .
6.3
g - 1 (x)
3
= x + 2 3
3
1
1
=
g (x)
3x - 2
. . . a straight line [ g (x) = 1 x + 2 ]
3
3
1
â g ⎛⎜ ⎞⎟ = 3 - 2
⎝ x⎠
x
NB: f -1 (x) g
1
f (x)
f - 1 is the inverse of f
(not of f(x) ).
. . . a hyperbola!
A1
2
x = 2y2
2
-1
1
1
g ⎛⎜ ⎞⎟ = 3 ⎛⎜ ⎞⎟ - 2
⎝ x⎠
⎝ x⎠
y = 2x
∴ 2y
3
. . . a hyperbola !
Equation of f :
∴ Equation of g :
â y = x + 2
+ 3)
x
O
g
â x + 2 = 3y
-1
4.2
2.5 f (2) = 0 (2 ; 0) lies on f
by inspection on the sketch
∴ Equation of g - 1 :
4.1
6
=1
2
Note: The domains and ranges are swopped.
3.4 Yes ; for both f and f -1 , each x-value is associated with
only one y-value. . . . f (x) = - 1 x + 2
y
2.3
f -1 (x)
f and
are reflections
in the line y = x.
So, swop x and y.
2
Range : y > 0 ; y ∈ » &
(b) Domain : x > 0 ; x ∈ » & Range : y ∈ » OR : Swop x and y in y = 2x + 6 and then make y the subject.
∴ y = -x +2
x +2 = -
. . . In Topic 4, you will convert this to y = log 2 x,
i.e. f -1 (x) = log 2 x, by using the definition of a log.
5.7 (a) Domain : x ∈ »
f -1
y = mx + c where m = + 1 & c = - 3
â y = 1 x-3
â Equation of f -1 : x = - 3y + 6
∴ 3y = - x + 6
f -1 (x)
...
(0; - 3) 3.3 Equation of f -1 :
2.2
y
[They are both one-to-one relations.]
For each value of x, there is only one y-value for both graphs.
&
5.5 x = 2
(0 ; - 3)
∴ 2x = y + 2
2.1 Domain : x ∈ » (6 ; 0)
x
5.6 Yes ; for every x-value, there is only one y-value.
∴ y = 2x - 2 1.2.4
(3 ; 0)
O
y = 1x +1
2
Equation of f :
1.2.3
y = 6
(put x = 0) â (0; 6) x
5.4 f is reflected in the line y = x to produce the image f -1 The rule : (x ; y) → (y ; x ) f -1
y-int. :
(a) f (x) = 1 x + 1 1.2.2
y=x
(0 ; 6)
(1 ; 0)
⎛ 1 ; -3 ⎞
⎜
⎟
⎝8
⎠
⎛1
⎞
⎜ ; - 3⎟ ⎝8
⎠
f
2x + 6 = 0
â C 1.2.1
y
3.2
y
f -1
8⎠
⎝
x=2
(8 ; 3)
⎛ -3; 1 ⎞ (0; 1)
⎜
⎟
That is why x and y are swopped to determine the
graph (and the equation) of the inverse of a function.
3.1
y=x
(3 ; 8)
• (2; 0) lies on f and (0; 2) lies on f -1
ANSWERS
x
2
∴ y
∴
= x
= x
2
y =±
x 2
6.4 No ; g is the inverse, but not the inverse function, because
it is not a function. It is a one-to-many relation. Note: A vertical line could cut g more than once.
Gr 12 Maths – Functions: Answers
8.2 y
6.5 (a) See the 2 ways in (1) and (2) below.
2
2
(1) Consider y = 2x ; x [ 0
(2) Consider y = 2x ; x m 0
y
y
O
2
â f -1 (x) =
y
y = 2x ; x m 0
- x â f -1 (x) = + x
2
2
y
y=x
f
O
8.7 (a)
y=x
x
(b)
(c)
(d) f - 1 (4) = 2 (4 ; 2) lies on f -1 Domain : x m 0 ; x ∈ » Range : y [ 0 ; y ∈ » Range : y m 0 ; y ∈ » & f (0) = - 2(- 3)(1) = 6
2
â E (0; 6) ∴
f -1 (x)
x
. . . i.e. y = + x
12.3
x =
2
The restriction on the values of x for f , i.e. x m 0, maps
The equation of f :
â The eqn. of the inverse of f :
y = -x
2
x = - y2
2
â y
. . . swop x & y
= - x . . . make y the subject
â y = ± - x ; but y m 0
â The equation of
f -1
onto the values of y for f -1 . (See the sketch above).
; xm0
x = (y + 2)2
∴ y+2 = ± x
(the inverse, which is a function) :
y
f
10. Equation of h - 1 :
y=x
∴ y =
- x
y =
- x
; y[0
x
f
2
Equation of g :
y = -1 +
x ; xm0
∴ Equation of g - 1 :
x = -1 +
y
11.
y
7.4 A decreasing function.
g-1
∴
-1
y=x
Copyright © The Answer
O
i.e. For x = 6, y = 2 or y = 0
13.1
Note: y m -1
because
x m0
g-1
-4
-2
0
-2 -
Find g - 1 by swopping
the coordinates of
points on g.
g
2-
h -1
2
4
6
x
Note: g and g - 1 are
symmetrical about
the line y = x.
-4 -
13.3
A2
y
4-
2
x
-3 < y ≤ 4 6-
y = x+1
-1
Domain : - 1 < x ≤ 6 Range :
∴ y = (x + 1) ; x m - 1 g
8.1 P′(0 ; 0), Q′(1 ; - 1), R′(4 ; 2) and S′(9 ; 3) No ; for each x-value in the domain there is not only one
y-value.
x
O
∴ y = x ; x[0 ∴ A . . . turning point of f is D(1; 8)
12.5
y=x
h
∴ x = y
Gradient of h
(8; 1) 13.2
2
O
x
∴ D y
...
3
12.4
f -1
x = - y
∴ Equation of h :
â D (1; 8) e.g. (6; 2) and (6; 0) both lie on f -1
O
-2
(a) m = - 6 = - 2 y=x
∴ y = ± x -2
But y [ - 2:
f -1
y
y = (x + 2) ; x [ - 2
∴ Equation of f -1 :
y = + -x 7.3
2
Equation of f :
9.
& f (1) = - 2(- 2)(2) = 8
â a = 6 Note: y = ± x, the inverse of y = x , is not a function.
y [ 0; y∈» roots
(b) a = f (2) = - 2(2 - 3)(2 + 1)
â a = (- 2)(- 1)(3)
8.9 Yes, f -1 is a function because for every x m 0 (the given
domain) there is only 1 value of y.
7.1 Domain of f : x m 0 ; x ∈ » 2
= 1
12.2 Q (3; 0) . . . f(3) = 0
= x
∴ y = ± x ; but y m 0
∴ y =
At D, x = -1 + 3 . . . ave. of
At E, x = 0
12.1
2
∴ Equation of f -1 : x = y
∴ y
2
Graph of y = (x + 1) ; x m - 1
&
f (2) = 4 (2 ; 4) lies on f Equation of f : y = x ; x m 0
8.8
2
Domain : x m 0 ; x ∈ » 7.2
-1
-1
f -1 (9)
For f -1 :
Range of f :
f (3) = 9 =3 x
g-1:
Graph of y = (x + 1)
f -1
(c) For f -1 :
& S′ (9; 3)
x
P(0 ; 0)
Graph of y = - 1 +
&
2
f -1
x
O
f
x
Graph of y =
8.6 P′ (0; 0), R′ (4; 2)
O
2
y = 2x ; x [ 0
(b) Eqn. of f :
-1
8.5 See sketch.
S ′ (9; 3)
R ′ (4; 2)
x
g:
8.4 y = x f -1
R(2; 4)
O
x
Note :
& S(3; 9) S(3; 9)
f
f
8.3 P(0; 0), R(2; 4)
f
& 8.5
g - 1 is a function because for every value of x, there is a
unique value of y.