unit 1 differential calculus - IGNOU

⎛ x 3 ⎞ ⎛ x 3 ⎞
fog (x) = f (g (x)) = f ⎜ − ⎟ = 2 ⎜ − ⎟ + 3 = x.
⎝ 2 2 ⎠ ⎝ 2 2 ⎠
Thus, we see that gof (x) = x and fog (x) = x are the identity functions on R. What
we have observed here is true for any two functions f and g which are inverses of
each other. Thus, if f : X → Y and g : Y → X are inverses of each other, then gof
and fog are identity functions. Since the domain of gof is X and that of fog is Y,
we can write this as :
gof = Ix, fog = Iy
This fact is often used to test whether two given functions are inverses of each
other or not.
1.3.7 Types of Functions
In this section, we shall talk about various types of functions, namely, even, odd,
increasing and decreasing. In each case, we shall also try to explain the concept
through graphs.
Even and Odd Functions
We shall first introduce two important classes of functions : even functions
and odd functions.
Consider the function f defined on R by setting
You will notice that
f (x) = x 2 , for all x ∈ R.
f (− x) = (− x) 2 = x 2 = f (x), for all x ∈ R.
This is an example of an even function. Let’s take a look at the graph
(Figure 1.11) of this function. We find that the graph (a parabola) is
symmetrical about the y-axis. If we fold the paper along the y-axis, we shall
see that the parts of the graph on both sides of the y-axis completely
coincide with each other. Such functions are called even functions. Thus, a
function f, defined on R is even if, for each x ∈ R, f (− x) = f (x).
y
0
Figure 1.11
The graph of an even function is symmetric with respect to the y-axis. We
also note that if the graph of a function is symmetric with respect to the
y-axis, the function must be an even function. Thus, if we are required to
draw the graph of an even function, we can use this property to our
advantage. We only need to draw that part of the graph which lies to the
right of the y-axis and then just take its reflection w. r. t. the y-axis to obtain
the part of the graph which lies on the left of the y-axis.
Now let us consider the function f defined by setting f (x) = x 3 , for all x ∈ R.
We observe that f (− x) = (− x) 3 = − x 3 = − f (x), for all x ∈ R. If we consider
x
Differential Calculus
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