Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
Functions and Graphs
1.
Find the distance and the coordinates of the midpoint of the line joining each pair
of points below.
a) A(4,8) and B(10,2)
b) P(4,6) and Q(-1, 8)
2.
Given M(4,-2), is the midpoint of P(2, 6) and Q(x, y). Find the value of x and y.
3.
Given points A, B and C are (-2,7), (6,-1) and (9,4) respectively and T is the
midpoint of AB. Find
a) coordinate T
b) the equation of TC
c) the equation of BC
4.
Find the equation of the straight line that passes through
a) A(2,3) and the gradient is 4
b) A(2, -4) and B(-5, 1)
c) R(1, 5) and is perpendicular to x + y + 15 = 0
d) C(2, 8) and is parallel to the line x + 2y + 15 = 0
e) the point of intersection between x + 2y = 8 and 2y – x =8 with the gradient of
6.
5.
In the triangle PQR, Q is the point (2, -3), R is the point (5, 3) and the equations
of the sides PQ, PR are x + y + 1 = 0 and 2x – y + 11 = 0 respectively. Find
a) the coordinates of P
b) the equation of the straight line passing through P and parallel to QR.
6.
The points of A, B, C and D are the vertices of a parallelogram. Given the sides of
AB and AD are 3x – 4y + 1 = 0 and 2x – y – 1 = 0 respectively and the
coordinates of C is (6,6). Find
a) the equations of BC and DC
b) the coordinates of the remaining vertices
c) the coordinates of F which is the intersection of AC and BD
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
7.
Find the equation of the straight line joining the point of intersection of the lines
x + 3y – 12 = 0 and 3x – y – 6 = 0 and perpendicular to the line 2x – y = 3.
8.
Find the coordinates R(x,y) which divide AB in the given ratio.
a) A(-1, 4) B(3, 1)
1:2 internally
b) A(3,5)
3:2 externally
B(-2,3)
9.
Determine whether the line 2x – 3y + 1 = 0 divide the straight line joining the
point of A(2, 1) and B(4, -1) internally or externally. Hence find the ratio of the
division.
10.
Given P(1,0), Q(4,3) and R(7,1) are vertices of a triangle. Find
a) the equation of PR
b) the distance from point Q to PR
c) the area of the triangle PQR
11.
If A ={1, 2, 3, 4} and B is set of all integers, draw the arrow diagram showing the
function f: x→x -3 where x A . What is the range of the function?
12.
Sketch the graph y = x2 -1 with domain -3  x  2. What is the range of the
relation?
13.
f(x) = x2 – 3x. Evaluate f(1), f(2) and f(-1).
14.
(x) = x2 – 8x + 7. Evaluate (0), (1). For what values of x is (x) = 0?
15.
For what values of x is the function (x) =
16.
For what values of x is the function (x – 5)(x – 7) negative? For what values of x
is the function ( x  5)( x  7) defined?
17.
Define y as an explicit function of x (if possible) when
a. x + 4y = x3
b. x + y + 2 = x2
c. x5 + y5 = 3.
d. y – x = 2
x
defined?
( x  1)( x  3)
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
18.
19.
If f(x) = 2x, g(x) = x – 1 and h(x) = x2, find
(a)
fg(x)
(b)
gh(x)
2
(d)
f g(x)
(e)
hgf(x)
(c)
(f)
fgh(x)
gh 2(x)
If f(x) = 2x – 3 and g(x) = x 2 + 5, find
(a)
gf(2)
(b)
fg(3)
(d)
gf(x)
(e)
fg(x + 1)
(c)
(f)
fg(a – 1)
g 2(x)
20.
If f(x) = 2x – 3 and fg(x) = 2x + 1, find g(x).
21.
If f(x) = x – 1 and gf(x) = 3 + 2x – x 2, find g(x).
22.
The function f and the composite function f  g are defined as follows:
f : x  x 2 + 1, x  0 and f o g : x  x2 - 2x + 2, x  1
Define the function g.
23.
The function g and the composite function g  f are defined as follows:
g : x  ln x, x  0 and g o f : x  2 ln(x + 1), x  -1
Define the function f.
24.
Find the inverse of each of the following functions, stating its domain.
(a)
f(x) = x 2 + 1,
x  0
2
(b)
f(x) = (x – 1) ,
x  1
(c)
g(x) = (x – 2)(x – 4),
x  3
2
(d)
h(x) =
,
x  3
x 3
x2
(e)
h(x) =
,
x  2
x2
25.
The functions f and g are defined by
f : x  2(x + 3)2 - 5,
x5
g:x
,
2
(a)
(b)
26.
x  -5
Show that f is a one-to-one function and find an expression for f –1(x).
Find an expression for gf(x).
The functions f and g are defined by
f : x  x3,
x R
g : x  2 - 3x,
x R
Find
(a)
(b)
x  -3;
fg(x)
(fg)-1(x)
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
27.
The functions f and g are defined by
1
f: x
ln x,
x  0
2
g:x
x  0
x,
(a)
(b)
28.
Find f -1 and state its domain.
Find the composite function g  f –1 and state its range.
The function f, g and h are defined as follows:
1
f : x  2 x  1, x 
2
2
g : x  x , x
h : x  ln x, x  0
Find these composite function (if exist)
h f
h  (g  f )
a)
d)
g f
g  (g  f )
b)
e)
gh
f  ( g  h)
c)
f)
29.
Sketch the graph of each of the following functions and its inverse.
(a)
f(x) = x + 3,
x  R
(b)
f(x) = 2x - 1,
x  R
(c)
f(x) = (x – 1)(x + 1),
x  0
3
(d)
f(x) = x2 + 3x - 4,
x  2
30. Find the inverse of each of the following functions. In each case, sketch the graph
of the function and its inverse.
(a) f(x) = x 3 - 3
(b) f(x) = 2 x
(c) f(x) = log (x + 1)
31. The function f is defined as
h : x  x 2  3x , x R .
Show that the function h is not one to one function. If the domain of h is modified to
{ x: x  R, x  L} so that the function is one to one, find the smallest values of L.
32. Find the domain and range of the following functions and sketch the graph.
(a) f(x) = 6 x
(b) f(x) = 3 x+1
(c) f(x) = e2x + 5
(d) f(x) = 1 – e2x
-2x
(e) f(x) = - e
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
33. Find the domain and range for the following functions, hence sketch the graph.
(a) f(x) = log5 x
(b) f(x) = log3 (-x)
(c) f(x) = ln(x-3)
(d) f(x) = -ln(x-3)
34. The function f is defined as
f : x  ln( x  1)
(a) Find f -1
(b) Determine the domain and range of f -1
(c) Sketch the graph f and f -1 on the same axis.
35. Given the function f(x) = e –x + 1.
(a) Find f -1 and determine the domain.
(b) Show that f -1 is a function.
(c) Sketch the graph f and f -1 on the same axis.
36. The function f and g is defined as follows
f : x  1 + 2e-x
 2 
g : x  ln 
, x  1
 x 1
(a) Find fg(x). What can we says about the function g.
(b) Sketch on the same axis the graph f and f -1.
37. The function f is defined as f : x  x  2  1, x  2. State the range of f.
Give the definition of f -1 which is equivalent. Sketch the graph of f and f -1 on the
same axis. State the relationship between the graph f and f -1.
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
PAST YEAR QUESTIONS
1. The functions f and g are defined as follows:
f : x  x2 -1 , x  R
g : x  e-x , x  R+
(a) Determine whether the function f is one to one. Explain your answer.
(b) Find g-1 and g-1 o f.
(c) Determine the domain of g-1.
(Mac 2001/7 marks)
2. Given f(x) = x2 + 1 and g(x) = x  1 .
(a) State the domains and ranges of f and g.
(b) Find f o g, g o f and f o g o f.
(c) On the same axes, sketch the graphs f and g.
(d) State the domain of f so that f is the inverse of g.
(Mac 2001/12 marks)
3. Given f(x) =
domain.
x , g(x) = x – 1 and h(x) = ex. Find f o g o h and determine the
(Mac 2002/7 marks)
4. The function f and g are given as follows
f(x) = x
g(x) = x – 1
(a) State the domain and range for f and g.
(b) Find f o g and sketch the graph.
(c) Does the f o g function one to one ? Give your reason.
(d) State the possible domain for f o g so that it has an inverse.
(e) Sketch the inverse of (f o g)-1.
(Mac 2002/12 marks)
5. Given that f(x) = 2x + 1 and h(x) = 2x2 + 4x + 1, find a function g such that
(fog)(x) = h(x).
Write g(x) in the form of a(x+b)2 + c, where a, b and c are constants.
(Mac 2003/6 marks)
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
6. Given that f , g and h as follows
f(x) = x
g(x) = x2 – 1
1
h(x) = , x  0
x
(a) Find F(x) = ( f o g o h)(x). State its domain and range.
(b) Find all the asymptotes of F and determine the interval where F is continuous.
(c) Find the values of x when F(x) = 5.
(d) Sketch the graph of F.
(Mac 2003/15marks)
7. Given g  x   3 x and h x  
1
.
x3
(a) Find f x  such that  f  g  h  x  
x
x 1
(b) Determine the domain of  f  g  hx .
(April 2004/6marks)
8. A function f is defined by
x
f x  
2
x 9
(a) State the domain of f .
(b) Find the vertical asymptotes.
(c) Determine lim f x  and lim f x  . Hence, state the horizontal asymptotes.
x
x
1
(d) Find f and determine the range of f .
(e) Sketch the graph of f .
(April 2004/15marks)
9. Functions f and g are defined as
f ( x)  e 2 x
Find f
1
,
g ( x )  1  x,
( x) and hence obtain ( g  f
1
x  R.
)( x).
(October2005/5 marks)
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
Answers
3
221 , ( ,1)
2
1.
a) 2 13 , (7,10) b)
2.
x = 6, y = -10
3.
a)(2,3)
4.
a)
4x – y – 5 = 0
b) 5x + 7y + 18 = 0
c)
x–y+4=0
d) x + 2y – 18 = 0
b) x – 7y + 19 = 0
c) 5x – 3y – 33 = 0
e) 6x – y + 4 = 0
b) 2x – y + 5 = 0
5.
a) P(-2, 11)
6.
a) BC : 2x – y – 6 = 0
DC : 3x – 4y + 6 = 0
7.
x + 2y – 9 = 0
8.
1
a) R ( ,3)
3
b) R(-12,-1)
9.
externally,
1:6
10.
a) x – 6y – 1 = 0
11.
range = { - 2, -1, 0, 1}
13.
- 2,- 2, 4
15.
all value except 1 and 3
17.
a )y 
18.
a. 2(x – 1) b. x2 – 1 c. 2(x2 -1) d. 4(x – 1)
19.
a. 6 b. 25 c. 2a2- 4a + 7 d. 4x2 -12x +14 e. 2x2 + 4x + 9
14.
b)
b) A(1,1), B(5,4), D(2,3)
15 37
37
12.
c)
15
2
range = { 8, 3, 0, - 1 }
x = 1 or x = 7
16.
i. 5 < x < 7
x3  x
b )y  x 2  x  2 c )y  3  x 5
4


1
21. 4 – x2 22. x – 1
ii. x  5  x  7
5
e. (2x – 1)2 f. x4 – 1
f. x4 + 10x + 30
20. x + 2
7 7
c) ( , )
2 2
23. f(x) = ( x + 1 )2
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
24 )a ). f 1 ( x )  x  1,x  1 D f 1  1,  
b ) f 1 ( x )  x  1,x  0
D
f
 0,  
-1
c )g 1 ( x )  x  1  3 Dg 1   1,  
2  3x
Dh1    ,0   0,  
x
2 1  x 
Dh1    ,1  1,  
e )h 1 ( x ) 
x 1
d )h 1 ( x ) 
26.a ) fg( x )  ( 2  3x )
3
25. b. g (f(x)) = x + 3

 x  e
ln( 2 x  1)
b)
d)
f)
c.
27.a ) f 1 ( x )  e 2 x
28.
a)
c)
e)
b ) g f 1
(ln x) 2
(2 x  1) 4
29. a.
f
x
b )  fg 
(2 x  1) 2
ln( 2 x  1) 2
2(ln x) 2  1
f
y=x
1
-1
-3
b.
f
y=x
f-1
1
1/2
-1
1
-1
1/2 1
-1
y=x
f-1
3
3
2x
(x)
2
range  0,  
f-1
-3
1
1
3
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
d.
f
y=x
f-1
-25/40 -3/2
-3/2
-25/40
ANSWERS FOR PAST YEAR QUESTIONS


1.
b.
g 1  f   ln x 2  1
2.
a.
Df  
J f  1, 
D g  1, 
J g  0, 
f g x
g f  x
b.
c.
Dg 1 x   0,
f  g  f  x2 1
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
c.
y
x
x
y=x
1
x
1
d.
3.
4.
J g  D f  0, 
D f  0, 
f  g  h  ex 1
a.
b.
Df  
J f  0,
Dg  
Jg  
 f  g ( x) 
x 1
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
f  g (x)
1
x
1
d.
i.
e.
i.
D f  g  1,
ii.
D f  g   ,1
for x  1
1
x
1
4.
e.
ii.
for x  1
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
f g
y=x
xxxx
1
1
f
5.
1
g ( x)  x 2  2 x
=  x  1  1
2
6.
 g
; x =R
a. F(x) = | 1/x2 – 1 | Domain F = ( -∞, 0) U ( 0, ∞ ),
Range F = [ 0, ∞ )
b. Asymptotes horizontal : y  1
Asymptotes vertical : x  0
Interval F continuous = D f   ,0  0,  
c. x  0.408  
1
6
d.
y
-1
1
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
7.
a. f  x  
1
1 x
b. Domain
8.
; x  1
 f  g  hx   ,1  1,0  0, 
a. Domain f   ,3  3, 
b. Vertical asymptotes : x  3
c. lim f x  1
x
lim
x
f x   1
Horizontal asymptotes : y  1
3x
d. f 1  x   
x2  1
Range f  Domain f 1   ,1  1,  
f x 
e.
y 1
0
x
y  1
x  3
9.
1
ln x, x  0
2
1
1  ln x, x  0
2
TIPS FOR TOPIC 6
x3
Mathematics QM 016
Topics 6: Functions and Graphs
________________________________________________________________________
1. Do not confuse composition with multiplication i.e. f o g(x) ≠ f(x) x g(x)
2. f o g means: do g first and then f . Do not get this the wrong way round.
3. When working with functions, f -1 means ‘the inverse of function f”, an not ‘the
reciprocal of function f ’
4. The graph of y = -f(x) is a reflection of the graph of y = f(x) in the x-axis
whereas
the graph of y = f(-x) is obtained by reflecting the graph of in the
y-axis.
5. For simple exponential function in the form of y = aebx+c +d , there is a horizontal
asymptote whereas for simple logarithmic function in the form of y = aln(bx +c)
+d, there is a vertical asymptote.