Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Exponential Functions: A Comprehensive Guide with Examples, Study Guides, Projects, Research of Elementary Mathematics

Because the exponent is unrestricted, the domain of an exponential function is all real numbers (−∞,∞). To find the range of an exponential.

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/27/2022

houhou
houhou 🇺🇸

4

(7)

269 documents

1 / 6

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
16-week Lesson 29 (8-week Lesson 23) Exponential Functions
1
Up until now we have worked with what are known as power functions,
which are functions of the form 𝑓(𝑥)= 𝑥𝑎. With a power function, the
base is a variable such as 𝑥 and the exponent 𝑎 is a constant such as 2.
Now we will work with exponential functions 𝑓(𝑥)= 𝑎𝑥, where the base
𝑎 is a constant, such as 2, and the exponent is a variable such as 𝑥. Keep
in mind that while we are inverting the base and the exponent when going
from power functions to exponential functions, those two types of
functions are NOT inverses of each other (more about this later).
Exponential Functions:
- 𝑓(𝑥)= 𝑎𝑥
o the exponent 𝑥 is a variable and base 𝑎 is a constant
o Examples:
𝑓(𝑥)= 2𝑥 𝑔(𝑥)= (1
3)𝑥
- there are no restrictions on the exponent 𝑥, but there are restrictions
on the base 𝑎
o 𝑎 cannot be negative (𝑎 0) because 𝑥 could be a fraction
(𝑥 = 1
2)
𝑎1
2 (or 𝑎 ) would be undefined if 𝑎 was negative
o 𝑎 cannot be zero (𝑎 0) because 𝑥 could be negative (𝑥 = −1)
𝑎−1 (or 1
𝑎) would be undefined if 𝑎 was zero
o 𝑎 cannot be one (𝑎 1) because 𝑓(𝑥)= 𝑎𝑥 would not be a
one-to-one function
𝑓(𝑥)= 1𝑥 is always equal 1, so it is not a 11 function
An exponential function 𝑓(𝑥)= 𝑎𝑥 must have a positive base other than 1
(𝑎 > 0 and 𝑎 1). Keep in mind that while the base of an exponential
function cannot be negative, an exponential function can be transformed
just like any other function, such as (𝑥)= −2𝑥, which is −1(2𝑥).
pf3
pf4
pf5

Partial preview of the text

Download Exponential Functions: A Comprehensive Guide with Examples and more Study Guides, Projects, Research Elementary Mathematics in PDF only on Docsity!

Up until now we have worked with what are known as power functions,

which are functions of the form 𝑓

𝑎

. With a power function, the

base is a variable such as 𝑥 and the exponent 𝑎 is a constant such as 2.

Now we will work with exponential functions 𝑓

𝑥

, where the base

𝑎 is a constant, such as 2 , and the exponent is a variable such as 𝑥. Keep

in mind that while we are inverting the base and the exponent when going

from power functions to exponential functions, those two types of

functions are NOT inverses of each other (more about this later).

Exponential Functions:

𝑥

o the exponent 𝑥 is a variable and base 𝑎 is a constant

o Examples:

𝑥

1

3

𝑥

  • there are no restrictions on the exponent 𝑥, but there are restrictions

on the base 𝑎

o 𝑎 cannot be negative

because 𝑥 could be a fraction

1

2

1

2

(or

𝑎 ) would be undefined if 𝑎 was negative

o 𝑎 cannot be zero

because 𝑥 could be negative

( 𝑥 = − 1

)

− 1

(or

1

𝑎

) would be undefined if 𝑎 was zero

o 𝑎 cannot be one

because 𝑓

𝑥

would not be a

one-to-one function

𝑥

is always equal 1 , so it is not a 1 − 1 function

An exponential function 𝑓

𝑥

must have a positive base other than 1

𝑎 > 0 and 𝑎 ≠ 1

. Keep in mind that while the base of an exponential

function cannot be negative, an exponential function can be transformed

just like any other function, such as ℎ

𝑥

, which is − 1

𝑥

Because the exponent 𝑥 is unrestricted, the domain of an exponential

function is all real numbers (−∞, ∞). To find the range of an exponential

function we can use an input/output table like the one given below.

Example 1: List the domain and range of 𝑓

𝑥

in interval notation.

Inputs Outputs

𝑥

2

− 5

5

− 4

4

− 3

3

− 2

2

− 1

0

1

2

3

4

5

2

The domain of 𝑓

𝑥

is all real numbers

and the range is

only positive numbers

. As shown in the table, anything can go in

for 𝑥, but only positive numbers come out.

Starting at the row

where 𝑥 = 0 and

𝑓

( 0

) = 1 , each of

the following rows

are just multiplied

by 10 :

1 ∙ 10 = 10

10 ∙ 10 = 100

100 ∙ 10 = 1000

So it should make

sense why the top

end of the range is

infinity; because

we are essentially

multiplying by 10

repeatedly when

our inputs are

consecutive

positive integers.

And going the

other direction in

the table, we end

up dividing by 10

repeatedly, which

is why we get

closer and closer

to zero. Keep in

mind that because

we are dividing by

10 repeatedly, we

will never reach

zero, or have

anything less than

zero. That is why

the range is ( 0 , ∞).

Example 3 : Given the exponential function ℎ

𝑥

, list the domain

and range in interval notation.

Keep in mind that the base of this exponential function is 3 , not − 3. So

𝑥

Inputs Outputs

𝑥

2

− 5

5

− 4

4

− 3

3

− 2

2

− 1

1

0

1

2

3

4

5

2

An alternative to making an input/output table for each of these problems

is to break an exponential function that has been transformed into smaller

pieces. For instance, the function 𝑓

𝑥

  • 8 could be broken up as

follows:

𝑥

only produces positive function values, so its range would be

𝑥

(or − 1 ∙ 9

𝑥

) would have nothing but negative function values,

because we would take all the positive function values from 9

𝑥

and negate

each of them, so the range of − 9

𝑥

would be

𝑥

  • 8 would take the outputs of − 9

𝑥

and simply add 8 to each

of them

, so its range would be

Again, this can be used as an alternative to making an input/output table

for each function. However if it is easier to identify the range of a

function by seeing the outputs in a table, feel free to stick with making

tables.

Example 4 : Given the exponential function ℎ(𝑥) = (

1

4

𝑥

, list the domain

and range in interval notation.

Example 5 : Given the exponential function ℎ

1

4

𝑥

, list the

domain and range in interval notation.

Example 6 : Given the exponential function 𝑘(𝑥) = − (

1

4

𝑥

− 4 , list the

domain and range in interval notation.