FINDING THE DOMAIN AND RANGE OF AN EXPONENTIAL FUNCTION

Exponential function will be in any of the following forms.

y = bx

(or)

y = abx

(or)

y = abx-h + k

Domain :

Domain of exponential function is all real values. (-∞, ∞).

Range :

(k, ∞) or (-∞, k) can be any one of them, based on exponential growth or decay can be decided.

Exponential growth or decay :

If b > 0 for growth and 0 < b < 1 when it is decay.

exponential-growth-decay-function.png

Horizontal asymptotes :

The exponential function which is in the form

y = abx-h + k

x = k is the horizontal asymptote.

Find:

1) Domain

2) Range

3) Exponential growth or decay

4) Horizontal asymptote

Problem 1 :

f(x) = 3x

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

The range is all positive real numbers.

(0, ∞)

Exponential growth or decay:

f(x) = 3x

if x = -3, f(x) = 3-3 = 1/27

if x = -2, f(x) = 3-2 = 1/9

if x = -1, f(x) = 3-1 = 1/3

if x = 0, f(x) = 30 = 1

if x = 1, f(x) = 31 = 3

Since the multiplication factor(b) is 3 > 1, this is an exponential growth function.

Horizontal asymptote:

y = 0

Problem 2 :

f(x) = -(3x)

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

The range is all negative real numbers.

(-∞, 0)

Exponential growth or decay:

f(x) = -(3x)

a = -1, b = 3 > 3

if x = -3, f(x) = -(3-3) = -1/27

if x = -2, f(x) = -(3-2) = -1/9

if x = -1, f(x) = -(3-1) = -1/3

if x = 0, f(x) = -(30) = -1

if x = 1, f(x) = -(31) = -3

Multiplication factor is 3, but we have negative. Then, it is reflection.

Horizontal asymptote:

y = 0

Problem 3 :

f(x) = 3-x

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

The range is all positive real numbers.

(0, ∞)

Exponential growth or decay:

f(x) = 3-x

if x = -3, f(x) = 3-(-3) = 27

if x = -2, f(x) = 3-(-2) = 9

if x = -1, f(x) = 3-(-1) = 3

if x = 0, f(x) = 30 = 1

if x = 1, f(x) = 3-(1) = 1/3

This is an exponential decay.

Horizontal asymptote:

y = 0

Problem 4 :

f(x) = (1/3)x

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

The range is all positive real numbers.

(0, ∞)

Exponential growth or decay:

a = 1, b = 1/3 < 1

f(x) = (1/3)x

if x = -3, f(x) = (1/3)-3 = 27

if x = -2, f(x) = (1/3)-2 = 9

if x = -1, f(x) = (1/3)-1 = 3 

if x = 0, f(x) = (1/3)0 = 1

if x = 1, f(x) = (1/3)1 = 1/3

This is an exponential decay function.

Horizontal asymptote:

y = 0

Problem 5 :

f(x) = 2x - 3

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(-3, ∞)

Exponential growth or decay:

a = 1, b = 2 > 1

if x = -3, f(x) = 2-3 - 3 = (1/8) - 3 = -2.875

if x = -2, f(x) = 2-2 - 3 = (1/4) - 3 = -2.75

if x = -1, f(x) = 2-1 - 3 = (1/2) - 3 = -2.5

if x = 0, f(x) = 20 - 3 = 1 - 3 = -2

if x = 1, f(x) = 21 - 3 = 2 - 3 = -1

if x = 2, f(x) = 22 - 3 = 4 - 3 = 1

This is exponential growth function.

Horizontal asymptote:

y = -3

Problem 6 :

f(x) = 2x-3

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(0, ∞)

Exponential growth or decay:

if x = -2, f(x) = 2-2-3 = 2-5 = 1/32

if x = -1, f(x) = 2-1-3 = 2-4 = 1/16

if x = 1, f(x) = 21-3 = 2-2 = 1/4       

if x = 2, f(x) = 22-3 = 2-1 = 1/2     

This is an exponential growth function.

Horizontal asymptote:

y = 0

Problem 7 :

f(x) = 2x+5 - 5

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(-5, ∞)

Exponential growth or decay:

if x = -2, f(x) = 2-2+5 - 5 = 23 - 5 = 3 

if x = -1, f(x) = 2-1+5 - 5 = 24- 5 = 11

if x = 0, f(x) = 20+5 - 5 = 25 - 5 = 27

if x = 1, f(x) = 21+5 - 5 = 26 - 5 = 59

This is an exponential growth function.

Horizontal asymptote:

y = -5

Problem 8 :

f(x) = -2-x

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(-∞, 0)

Exponential growth or decay:

if x = -2, f(x) = -2-(-2) = -22 = -4

if x = -1, f(x) = -2-(-1) = -21 = -2

if x = 0, f(x) = -2(0) = -1

if x = 1, f(x) = -2-1 = -1/2

if x = 2, f(x) = -2-2 = -1/4

This is an exponential growth function.

Horizontal asymptote:

y = 0

Problem 9 :

f(x) = -2x+3 + 1

Solution:                         

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(-∞, 1)

Exponential growth or decay:

if x = -2, f(x) = -2-2+3 + 1 = -2 + 1 = -1

if x = -1, f(x) = -2-1+3 + 1 = -4 + 1 = -3

if x = 0, f(x) = -20+3 + 1 = -8 + 1 = -7

if x = 1, f(x) = -21+3 + 1 = -16 + 1 = -15

if x = 2, f(x) = -22+3 + 1 = -32 + 1 = -31

This is an exponential decay function.

Horizontal asymptote:

y = 1

Problem 10 :

f(x) = (1/2)x-3 - 4

Solution:

Domain:

The domain is all real number.

(-∞, ∞)

Range:

(-4, ∞)

Exponential growth or decay:

if x = -2, f(x) = (1/2)-2-3 - 4 = 32 - 4 = 28

if x = -1, f(x) = (1/2)-1-3 - 4 = 16 - 4 = 12

if x = 0, f(x) = (1/2)0-3 - 4 =8 - 4 = 4

if x = 1, f(x) = (1/2)1-3 - 4 = 4 - 4 = 0

if x = 2, f(x) = (1/2)2-3 - 4 = 2 - 4 = -2

This is an exponential decay function.

Horizontal asymptote:

y = -4

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