Exponential Functions

MM3A2e Investigate characteristics: domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rate of change.

An exponential function is a function of the form

where a is a positive real number (a > 0) and . The domain of f is the set of all real numbers.

3 2 1 0 1 2 3

2

4

6

(0, 1)

(1, 3)

(1, 6)

(-1, 1/3) (-1, 1/6)

Summary of the characteristics of the graph of

a >1

• The domain is all real numbers. Range is set of positive numbers.

• No x-intercepts; y-intercept is 1.

• The x-axis (y=0) is a horizontal asymptote as

• a>1, is an increasing function and is one-to-one.

• The graph contains the points (0,1); (1,a), and (-1, 1/a).

3 2 1 0 1 2 3

2

4

6

(-1, 3)

(-1, 6)

(0, 1) (1, 1/3) (1, 1/6)

Summary of the characteristics of the graph of

0 <a <1

• The domain is all real numbers. Range is set of positive numbers.

• No x-intercepts; y-intercept is 1.• The x-axis (y=0) is a horizontal asymptote as

• 0<a<1, is a decreasing function and is one-to-one.

• The graph contains the points (0,1); (1,a), and (-1, 1/a).•The graph is smooth continuous with no corners or gaps.

0

5

10

(0, 1)

(1, 3)

y x3

0

5

10

(0, 1)(-1, 3)

y x 3

0

5

10

(0, 3)

(-1, 5)

y = 2

y x 3 2

Horizontal Asymptote: y = 2

Range: { y | y >2 } or

Domain: All real numbers

3 2 1 0 1 2 3

2

4

6

y x2

y x3 y ex

Characteristics of Exponential Functions f(x) = ax

• Domain: (-∞,∞)

• y-intercept is the point (0,1)

• Continuous

• Increasing if a > 1

• Range: (0,∞)

• No x-intercept

• Horizontal asymptote y=0

• Decreasing if a < 1