Exponential Functions

Exponential Functions

The exponential function with base b, where b > 0, has the following form:

f(x) = b^x
-¥ < x < ¥

where b is a constant. In an exponential function, the base of f(x) is a constant real number, while the exponent is the dependent variable of the function.

The graphs of all exponential functions (with a positive base) share similar properties:

(1) Since the value of b^x is positive for b > 0, the graph of the exponential function

f(x) = b^x

lies above the x-axis, but never touches it, as the value of b^x can never be zero.

(2) When b > 1, the value of b^x increases towards infinity as x approaches infinity, while it decreases to (but does not become) zero as x approaches negative infinity. Therefore, the graph of f(x) = b^x curves to the right and moves upward infinitely as x ® ¥, while it approaches the x-axis asymptotically as x ® -¥.

When 0 < b < 1, the value of b^x decreases to (without reaching) zero as x approaches infinity, while it increases towards infinity as x approaches negative infinity. Therefore, the graph of f(x) = b^x approaches the x-axis asymptotically as x ® ¥, while it curves to the right and moves upward infinitely as x ® -¥.

When b = 1, the value of b^x is always equal to 1, since 1^x = 1 for any value of x. Thus, the graph of f(x) = b^x is the ordinate y = 1.

(3) Since b⁰ = 1 for any real number b, the graph of the exponential function

f(x) = b^x

for any real number b, includes the point (0,1).

(4) For b > 0,

(1/b)^x = (b^-1)^x = b^-x

Thus, the graphs of f(x) = (1/b)^x and f(x) = b^x are reflections of each other through the y-axis.

EX. The exponential function

f(x) = 3^x

contains the following points:

 

x	3x	(x, y)
-2	1/9	(-2, 1/9)
-1	1/3	(-1, 1/3)
0	1	(0, 1)
1	3	(1, 3)
2	9	(2, 9)

EX. The exponential function

contains the following

x	(1/3)x	(x, y)
-2	9	(-2, 9)
-1	3	(-1, 3)
0	1	(0, 1)
1	1/3	(1, 1/3)
2	1/9	(2, 1/9)

Note that the graphs of f(x) = 3^x and f(x) = (1/3)^x are reflections of each other across the y-axis.

The quantity b^x, where b > 0 and b ¹ 1, obey the following axioms:

(1) b^x = b^y if and only if x = y.
(2) If b > 1 and r < s < t, then b^r < b^s < b^t
If 0 < b < 1 and r < s < t, then b^r > b^s > b^t

EX.
3^x = 27
3^x = 3³
x = 3

4^x = 128
(2²)^x = 2⁷
(2)^2x = 2⁷
2x = 7
x = 3.5

2^x < 2^y
x < y

(0.25)^m > (0.25)ⁿ
m < n