Graphing a More Complicated Exponential FunctionGraph
Find some ordered pairs. We let and and find values of or y.
Let LetThese ordered pairs, and along with the other ordered pairs shown in
the table, lead to the graph in FIGURE 7. The graph is similar to the graph of
except that it is shifted to the right and rises more rapidly.
ƒ 1 x 2 = 3 x
A0, 811 B^1 2, 1^2 ,
y= 3 -^4 , or y= 30 , or 1
1
81
y= 32102 -^4 x=0. y= 32122 -^4 x=2.
x= 0 x= 2 ƒ 1 x 2 ,
ƒ 1 x 2 = 32 x-^4.
EXAMPLE 3
582 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions
Characteristics of the Graph of1. The graph contains the point
2. The function is one-to-one. When the graph will risefrom left to
right. (See FIGURE 5.) When the graph will fallfrom left to right.
(See FIGURE 6.) In both cases, the graph goes from the second quadrant to the
first.
3. The graph will approach the x-axis, but never touch it. (From Section 7.4,
recall that such a line is called an asymptote.)
4. The domain is 1 - q, q 2 ,and the range is 1 0, q 2.
06 a 6 1,
a 7 1,
1 0, 1 2.
ƒ 1 x 2 ax46820 123f (x) = 3^2 x–4xyFIGURE 7xy
0
1
21
391
91
81NOW TRYOBJECTIVE 3 Solve exponential equations of the form for x.
Until this chapter, we have solved only equations that had the variable as a base, like
. In these equations, all exponents have been constants. An exponential equa-
tionis an equation that has a variable in an exponent, such as
We can use the following property to solve certain exponential equations.
9 x=27.
x^2 = 8
axak
Property for Solving an Exponential EquationFor and a 70 aZ1, if axay then xy.
This property would not necessarily be true if a=1.
NOW TRY
EXERCISE 3
Graph ƒ 1 x 2 = 42 x-^1.
NOW TRY ANSWER
3.
xy4
–1 0 1f(x) = 4^2 x – 1