Intermediate Algebra (11th edition)

580 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing win- dow. See the Connections box.

45.ƒ 1 x 2 =x^3 + 5 46.ƒ 1 x 2 = 23 x+ 2

ƒ 1 x 2 = 2 x- 7 ƒ 1 x 2 =- 3 x+ 2

If find each value indicated. In Exercise 50, use a calculator, and give the answer to the nearest hundredth. See Section 3.6.

49.ƒa- 50.ƒ 1 2.73 2

1

2

ƒa b

1

2

ƒ 132 b

ƒ 1 x 2 = 4 x,

PREVIEW EXERCISES

OBJECTIVES OBJECTIVE 1 Define an exponential function.In Section 8.2we showed

how to evaluate for rational values of x.

and

Examples of for rational x

In more advanced courses it is shown that exists for all real number values of x,

both rational and irrational. The following definition of an exponential function

assumes that axexists for all real numbers x.

2 x

2 x

2 -^1 = 2 1/2 = 22 , 2 3/4= 2423 = 248

1

2

23 =8, ,

2 x

Exponential Functions

10.2

1 Define an exponential function. 2 Graph an exponential function. 3 Solve exponential equations of the form for x. 4 Use exponential functions in applications involving growth or decay.

ax=ak Exponential Function

For and all real numbers x,

defines the exponential function with base a.

ƒ 1 x 2 ax

a 7 0,aZ1,

NOTE The two restrictions on the value of a in the definition of an exponential

function are important.

1.The restriction is necessary so that the function can be defined for all real

numbers x. Letting abe negative ( for instance) and letting would

give the expression which is not real.

2.The restriction is necessary because 1 raised to any power is equal to 1,

resulting in the linear function defined by ƒ 1 x 2 = 1.

aZ 1

1 - 22 1/2,

a=-2, x=^12

a 70

ƒ(x)ax

OBJECTIVE 2 Graph an exponential function.When graphing an exponen-

tial function of the form pay particular attention to whether or

06 a 6 1.

ƒ 1 x 2 = ax, a 71