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

...

...

                                       graph-of-an-exponential-function

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

                                                   graph-of-an-exponential-function-case-2

(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 = ax 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.

                                                  graph-of-an-exponential-function-case3

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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