Higher Engineering Mathematics

FUNCTIONS AND THEIR CURVES 199

C

Now try the following exercise.

Exercise 85 Further problems on simple

transformations with curve sketching

Sketch the following graphs, showing relevant

points:

(Answers on page 213, Fig. 19.39)

1.y= 3 x− 5

2.y=− 3 x+ 4

3.y=x^2 + 3

4.y=(x−3)^2

5.y=(x−4)^2 + 2

6.y=x−x^2

7.y=x^3 + 2

8.y= 1 +2 cos 3x

9.y= 3 −2 sin

(

x+

π

4

)

10.y=2lnx

19.3 Periodic functions

A functionf(x) is said to beperiodiciff(x+T)=
f(x) for all values ofx, whereTis some positive
number.Tis the interval between two successive rep-
etitions and is called the period of the functionf(x).
For example,y=sinxis periodic inxwith period
2 πsince sinx=sin (x+ 2 π)=sin (x+ 4 π), and so
on. Similarly,y=cosxis a periodic function with
period 2πsince cosx=cos (x+ 2 π)=cos (x+ 4 π),
and so on. In general, ify=sinωtory=cosωtthen
the period of the waveform is 2π/ω. The function
shown in Fig. 19.24 is also periodic of period 2π
and is defined by:

f(x)=

{

−1, when−π≤x≤ 0

1, when 0≤x≤π

f(x)

0

1

− 1

− 2 ππ−π 2 π x

Figure 19.24

19.4 Continuous and discontinuous

functions

If a graph of a function has no sudden jumps or

breaks it is called acontinuous function, examples

being the graphs of sine and cosine functions. How-

ever, other graphs make finite jumps at a point or

points in the interval. The square wave shown in

Fig. 19.24 hasfinite discontinuitiesasx=π,2π,

3 π, and so on, and is therefore a discontinuous func-

tion.y=tanxis another example of a discontinuous

function.

19.5 Even and odd functions

Even functions

A functiony=f(x) is said to be even iff(−x)=f(x)

for all values ofx. Graphs of even functions are

always symmetrical about they-axis (i.e. is a mirror

image). Two examples of even functions arey=x^2

andy=cosxas shown in Fig. 19.25.

− 3 − 2 − 10 1 2 3x

2

4

6

8

y

y = x^2

(a)

−π 0 π/2 π x

y

y = cosx

(b)

−π/2

Figure 19.25

Odd functions

A functiony=f(x) is said to be odd iff(−x)=−f(x)

for all values ofx. Graphs of odd functions are

always symmetrical about the origin. Two examples