Exponential Functions

1. Exponential Functions

2. 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. (1, 6) (1, 3) (-1, 1/6) (-1, 1/3) (0, 1)

4. 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).

5. (-1, 6) (-1, 3) (0, 1) (1, 1/3) (1, 1/6)

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.

8. (1, 3) (0, 1)

9. (-1, 3) (0, 1)

10. (-1, 5) (0, 3) y = 2

11. Domain:All real numbers Range: { y | y >2 } or Horizontal Asymptote:y = 2

14. The number e is defined as the number that the expression In calculus this expression is expressed using limit notation as