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# Exponential functions

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Exponential Function Review

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### Exponential functions

1. 1. Digital LessonExponential Functions and Their Graphs
2. 2. The exponential function f with base a isdefined by f(x) = axwhere a > 0, a ≠ 1, and x is any real number.For instance, f(x) = 3x and g(x) = 0.5xare exponential functions. 2
3. 3. The value of f(x) = 3x when x = 2 is f(2) = 32 = 9The value of f(x) = 3x when x = –2 is 1 f(–2) = 3 –2 = 9The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625 3
4. 4. The graph of f(x) = ax, a > 1 y 4 Range: (0, ∞) (0, 1) x 4 Horizontal Asymptote y=0Domain: (–∞, ∞) 4
5. 5. The graph of f(x) = ax, 0 < a < 1 y 4 Range: (0, ∞)Horizontal Asymptote y=0 (0, 1) x 4 Domain: (–∞, ∞) 5
6. 6. Example: Sketch the graph of f(x) = 2x.x f(x) (x, f(x)) y-2 ¼ (-2, ¼) 4-1 ½ (-1, ½) 20 1 (0, 1)1 2 (1, 2) x2 4 (2, 4) –2 2 6
7. 7. Example: Sketch the graph of g(x) = 2x – 1. State thedomain and range. y f(x) = 2xThe graph of thisfunction is a verticaltranslation of the 4graph of f(x) = 2xdown one unit . 2Domain: (–∞, ∞) x y = –1Range: (–1, ∞) 7
8. 8. Example: Sketch the graph of g(x) = 2-x. State thedomain and range. y f(x) = 2xThe graph of thisfunction is areflection the graph 4of f(x) = 2x in the y-axis.Domain: (–∞, ∞) x –2 2Range: (0, ∞) 8
9. 9. The graph of f(x) = ex y x f(x) 6 -2 0.14 -1 0.38 4 0 1 2 1 2.72 2 7.39 x–2 2 9
10. 10. The irrational number e, where e ≈ 2.718281828…is used in applications involving growth anddecay.Using techniques of calculus, it can be shownthat n  1  1 +  → e as n → ∞  n 10