**Example** Sketch the graph of following

f( x ) = 2x and g( x ) = ( 1 /2)^{x}

**Solution**

Let's firstly make a table of values for these two functions.

Following is the sketch of both of these functions.

This graph shows some very nice properties regarding exponential functions in general.

**Properties of f( x ) = b **^{x}

1. f(0)= 1. The function will take the value of 1 at x=0 always

2. f( x ) ≠ 0 . An exponential function will never get zero.

3. f( x ) > 0 . An exponential function is always get positive.

4. The previous two properties can be summarized through saying that the range of an exponential function is (0, ∞ ) .

5. The domain of exponential function is [ -∞, ∞ ] . In other terms, you can plug every x in an exponential function.

6. If 0 < b < 1 then,

a. f ( x ) → 0 as x → ∞

b. f ( x ) → ∞ as x → -∞

7. If b >1 then,

a. f (x ) → ∞ as x → ∞

b. f ( x ) → 0 as x → -∞

These will all be extremely useful properties to remember .