Exponential Function

Exponential and Logarithmic Functions

The function f(x) = ax, a > 0 where the base 'a' is constant and index x is a variable, is called an exponential function.

Clearly, x ε R so domain of f(x) is R and for no value of x, f(x) < 0 so range of 'f' is R - (-∞, 0] or (0, ∞)

Graph of an exponential function: y = ax:

The graph is different for 0 < a < 1 and a > 1, so we will discuss these cases separately.

Case I.     a > 1

Let a = 2. The domain is [-∞, ∞].

The value table is as given below

X	...	-3	-2	-1	0	1	2	3	4	5	6	...	...
f(x)	...	1/8	¼	½	1	2	4	8	16	32	64	...	...

Note:

(i)     The curve approaches x-axis as x → -∞

        So x-axis i.e. line y = 0 is the asymptote of y = ax. for a > 1

(ii)    This function is increasing strictly as x increases.

        So, it is a strictly increasing function, hence invertible.

Case II:            0 < a < 1

Let a = 1/2  Domain of f is (-∞, ∞) The value table is as under

X	-3	-2	-1	0	1	2	3
f(x)	8	4	2	1	½	¼	1/8

We observe that

(i) As x becomes very large, f(x) approaches x axis

 i.e. y = 0 is the asymptote of f(x) for a < 1

(ii) y = a^x decreases strictly as x increases for 0 < a < 1

So it is a strictly decreasing function. Hence, y = ax is a monotonic function for any a ≠ 1.

For a < 0 the exponential function in not defined precisely and for a = 1 it turns out to be constant function.

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