Exponential Functions
Exponential Functions
The exponential function with base b, where b > 0, has the following form:
f(x) = bx
-¥ < 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 bx is positive for b > 0, the graph of the exponential function
f(x) = bx
lies above the x-axis, but never touches it, as the value of bx can never be zero.
(2) When b > 1, the value of bx 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) = bx 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 bx 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) = bx 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 bx is always equal to 1, since 1x = 1 for any value of x. Thus, the graph of f(x) = bx is the ordinate y = 1.
(3) Since b0 = 1 for any real number b, the graph of the exponential function
f(x) = bx
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) = bx are reflections of each other through the y-axis.
EX. The exponential function
f(x) = 3x
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) = 3x and f(x) = (1/3)x are reflections of each other across the y-axis.
The quantity bx, where b > 0 and b ¹ 1, obey the following axioms:
(1) bx = by if and only if x = y.
(2) If b > 1 and r < s < t, then br < bs < bt
If 0 < b < 1 and r < s < t, then br > bs > bt
EX.
3x = 27
3x = 33
x = 3
4x = 128
(22)x = 27
(2)2x = 27
2x = 7
x = 3.5
2x < 2y
x < y
(0.25)m > (0.25)n
m < n