1.5 – Exponential Functions

Basic Definitions

)times(... naaaan

nn

aa

1

mnn mnm aaa /

Exponential Function

xaxf )(

where a > 0 and a ≠ 1.

kacxf hx )(

With translations:

Natural Exponential Function

xexf )(

where e is the natural base and e 2.718…

Try This

Function f(x) = 2x h(x) = (0.5)x g(x) = ex

Domain

Range

Increasing or Decreasing

Point Shared On All Graphs

(-∞, ∞) (-∞, ∞) (-∞, ∞)

(0, ∞) (0, ∞) (0, ∞)

Inc. Dec. Inc.

(0, 1)

Properties of f (x) = ax

Domain: Range:Increasing for:Decreasing for:Point Shared On All Graphs:Asymptote:

(-∞, ∞)(0, ∞)

a > 10 < a < 1

(0, 1)y = 0

Graph Sketching

Use translation of functions to graph the following. Determine the domain and range of each.

1. f (x) = -5(x + 2) – 3

2. g(x) = (1 / 3)(x – 1) + 2

Try This

Determine the values of x for which 2x > x 4. Write your answer using interval notation. You may use your graphing calculator.

Try This

An isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after 60 hours.

(b) Find the amount remaining after t hours.

(c) Estimate the amount remaining after 4 days.

(d) Use a graph to estimate the time required for the mass to be reduced to 0.01 g.