|
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
|
