Composition of Functions
End of block exercises
Exercise (1)
f ( x )  x  9, g ( x )  3  x 2 , and , h( x )  1  2 x
find
a ) g ( f ( x ))
b)h( g ( f ( x )))
c) f (h( g ( x )))
d ) f ( f ( x ))
Exercise (2)
If
1
f ( x) 
x
find
f ( f ( x )) 
Exercise (3)
If
y ( x )  x  3, and , z( x ) 
2
find
y ( z( x )) 
and
z( y ( x )) 
x
Exercise (6)
If
r ( t )  2t  3
and
s( t )  7t 2  t
find
r ( s( t )) 
and
s( r ( t )) 
Exercise (7)
If
f ( x) 
1
1
, and , g ( x ) 
x3
x3
find
f ( g ( x )) 
and
g ( f ( x )) 
and
f ( f ( x )) 
Exercise (8)
If
2
f ( x )  , and , g ( x )  7  x
x
find
g ( f ( x )) 
What is the domain of g ( f ( x )) ?
Inverse Functions


Exercise (1)
Find the inverse of
each of the following
functions
f ( x)  4 x  7
f ( x)  x
f ( x )  23x
1
f ( x) 
x 1
1
f ( x) 
( 4 x  3)
7
1
f ( x) 
x 1
2
f ( x)  5  4 x
and show that
f (f

( x ))  x
4t  3
5
x 1
g( x) 
x 1
f (t ) 
Parametric Representation of a
function

y= f (x) is called the Cartesian form, An
alternative representation is to write
expressions for y and x in terms of a third
variable known as a parameter, such as y (t),
x (t).
Parametric Representation of a
function




End of block exercises
1) Consider the
parametric equation
A) Plot a graph of this
function
B) Find an explicit
expression for y in
terms of x.
x  3t , y  9t
2
Parametric Representation of a
function


2) Given the parametric
equations
x  3t  2,
Plot a graph of y
against x.
y  3t  5
Parametric Representation of a
function

3) Obtain the Cartesian
equation of the function
defined parametrically
by
x  t , y  1  t , for 0  t  1
Parametric Representation of a
function

4) Plot a graph of the
function define by
x   t , y  t , for t  0
Describing Functions


Continuous and Discontinuous functions
The limit of function
Describing Functions


Exercises
2) Study graphs of the
functions
2
y  x , and y   x

Are these continuous
functions?
2
Describing Functions



3) Study graphs of
Are these continuous
functions?
4)Draw a graph of the
function (see a book for
limits).
2 x  1 x  3

f ( x )  5
x3
6
x3

y  3x  2, and y  7 x  1
Periodic functions

Any function that has a definite pattern
repeated at regular intervals is said to be
periodic, ( T period).
Odd and even functions



Even function: any function
that is symmetrical about the
vertical axis.
An even function is such
that for all x.
f ( x)  f ( x)
Odd and even functions


Odd function: any
function that possesses
rotational symmetrythat is, the graph on the
right can be obtained
by rotating the curve on
the left through 180º
about the origin.
An odd function is such
that
f (  x)   f ( x)
Odd and even functions


Exercises:
1) Classify the following as odd, even
or neither. If necessary sketch a
graph to help you decide
a) f ( x)  6, b) f ( x)  x , c) f ( x)  2 x  1
d ) f ( x)  x, e) f ( x)  2 x
2
End of block exercises

1) Sketch a graph of the function
x 2
x  0

f ( x)   x  2
0 x  2
4
x  2

Find
a ) lim x  0  f ( x )
b) lim x  0  f ( x )
c) lim x  0
f ( x)
d ) lim x  2  f ( x )
e) lim x  2  f ( x )
f ) lim x  2 f ( x )
End of block exercises

2) A function is periodic with period 2 and is
even. Sketch a possible form of this function.

3) A function is periodic with period 1 and is
odd. Sketch a possible form of this function.
The straight line

All linear functions can be written in the form
 f(x)=ax+b
 Or
 y=ax+b
The straight line


Exercise:
State which of the following functions will
have straight line graphs:
1
a) y( x)  3x  3 b) f (t )  t c) f ( x) 
x
d ) g( x)  13
e) f (t )  2  t
2
The straight line


In the linear function y= ax+b
a is the gradient of the graph and b is its
vertical intercept,
The straight line


Exercise:
For each of the following, identify the
gradient and vertical intercept:
a ) f ( x )  2 x  1 b ) f ( t )  3 c) g ( t )   2 t
d ) y( x)  7  17 x e) f ( x)  mx  c
The straight line

The gradient of the line joining A(x1,y1) and
B(x2,y2) is given by
vertical dis tan ce
y2  y1
gradient 

horizontal dis tan ce x2  x1
The straight line

Exercise:

1) Calculate the gradient of the line joining
(1,0) and (15,-3).
2) Calculate the gradient of the line joining
(10,-3) and (15,-3)

The straight line

The line passing through points A(x1,y1)
and B(x2,y2) is given by
y  y1 y 2  y1

x  x1 x 2  x1
The straight line

The distance between points A(x1,y1),
B(x2,y2) is given by
dis tan ce   x2  x1    y2  y1 
2
2
The straight line




Exercise:
1) Find the equation of the line joining (1,5)
and (-9,2).
2) Find the gradient and vertical intercept of
the line joining (8,1) and (-2,-3).
3) Find the distance between the points (4,5)
and (-17,1).
The straight line


End of block exercises:
1) State the gradient and vertical intercept of
a ) y  8 x  3 b) y  3t  2 c) y  9
d ) y  3t
e) f ( x )   3  4 x
The straight line




3) Find the equation of the line that passes
through A(0,3) and B(11,-1)
4) Find the gradient of the line that passes
through A(-9,1) and B(2,16).
5) Find the distance between the points with
coordinates (9,1) and (12,1).
6) Find the distance between the points with
coordinates (19,-2) and (-12,1).
The straight line



7) The pointsA(x1,y1) lies on the line
y=-2x+3. If the value of x1 is increased by 7,
what is the resulting change in the value of
y1.
8) Find the equation of the line passing
through A(2,-1) and B(5,8).
Common engineering functions

Polynomial functions:
A polynomial expressions has the form
n 1
n 2
an x  aa1 x  aa2 x ...a2 x  a1 x  a0
n
Where n is anon-negative integer.
2
1
Common engineering functions

A polynomial function has the form
n 1
n 2
p( x)  an x  aa1 x  aa2 x ...a2 x  a1 x  a0
n
2
1
Common engineering functions

Rational functions has the form
P( x )
R( x ) 
Q( x )


Where P and Q are polynomial expressions: P is called
the numerator and Q is called the denominator.
The Poles of a rational function are any values that makes
the denominator zero.
Common engineering functions

The modulus function is defined as
f ( x)  x
Common engineering functions

The Unit step function
is defined as follows
1 t  0
u( t )  
0 t  0
Common engineering functions

The delta function, or unit impulse function
 (t )
Common engineering functions

Exercises:
1) If G ( s) 
2) If
F ( s) 
3) If
F ( s) 
4) If
F (t ) 
5) If
f ( x) 
6) If G ( s) 
7) If
s
find G ( s   )
s2  w2
w
find F ( s   )
2
2
s  w
1
find 5F (5s)
s1
1
find sF ( s)
t 1
3x 2 find f ( t   )
1
s1
f ( x )  3x 2
find G ( jw)
find
f (
15
t)
2
Common engineering functions






8) The signum function is defined as form
below.
a) Sketch a graph of this function
b) Is this function discontinuous or continuous?
c) Is this function odd, even, or neither?
d) Is this function periodic?
e) Is this function many-to-one or one-to-one?
 1 x  0 


f ( x)  sgn( x)   1 x  0 
0

x

0


Common engineering functions

9) a) Sketch a graph of the function
u(t  1)  u(t  2)


b) State the position of any discontinuous.
Common engineering functions




10) The ramp function is defined as below
a) Sketch a graph of this function
b) State the position of any discontinuous.
c) Find
lim x 0 f ( x )
0
f ( x)  
kx
x0
x0
Common engineering functions

11) Sketch a graph of
y x x

12) Sketch a graph of
f ( x)  u( x  1) x
Common engineering functions

13) State the poles of the following rational
functions:
1
a ) F ( s) 
s
s 2  2s  3
b) F ( s) 
( s  1) 3
1
c) F ( s) 
3s  2
Common engineering functions

14) Consider the function
f ( x)  5  4 x
Find
f 1 ( x), Show that f ( f 1 ( x))  f 1 ( f ( x))  x
Common engineering functions

15) Find the inverse of the function
1
f ( x )  (4 x  3)
7
Common engineering functions

16)
If
f ( x )  x 2  3x
and
g( x)  x  2
find
a ) f ( f ( x ))
b) f ( g ( x ))
c) g ( f ( x ))
d ) g ( g ( x ))
Common engineering functions

17) State the rule that describes the function
y  3( x  1)
2
Common engineering functions


18) Write a formula for the function given by
the rule ‘subtract the cube of the input from
the square of the input’.
19) State the domain and range of the
function
y  3t  17, 0  t  9
Common engineering functions

20) The maximal domain of a function is the
largest possible domain that can be defined
for that function. Find the maximal domain of
the function
5
f (t ) 
t
Common engineering functions

21) Find the inverse of the function
3
f ( x) 
x

22) Find the equation of the straight line
passing through (-1,4) and (-4,1). Does the
line pass through(-2,3)?
Common engineering functions

Electrical Engineering and Electronics- Reactance of a
capacitor. The reactance of a capacitor is its
resistance to the passage of alternating current.
Reactance, X, measured in ohms, is given by
1
X 
2fc


Where f is the frequency of the current in Hertz and C
is the capacitance, measured in farads. Note that X is
a function of f. Calculate the reactance when the
frequency is 50 Hertz and the capacitance is
6
Farads.
10
Common engineering functions

26) Extension of a spring. A spring has a
natural length of 90cm. When a 1.5kg mass
is suspended from the spring, the length
extends to 115cm.Calculate the length when
a 2.5kg mass is suspended from the spring.
Common engineering functions

27) A curve is defined parametrically by
x  t  1, y  t  1
2
Obtain y explicitly in terms of x.