9.2 Exponential Functions

A function that can be expressed in the form f (x) = b x, b > 0 and b ≠ 1 is called an exponential function Domain =  Range = y > 0 (positive Real #s) To get an idea of their basic shape, let’s graph 2 using a table of values: Ex 1) a) h(x) = 2 x xy –1 ¼ 2 –1 = = –2 2 –2 = = 2 2 = 4 ½ = 8 xy –1 1/25 01 – / b) g(x) = 5 x D: , R: y > 0,x-int: none y-int: (0, 1),increasing, one-to-one  same 26

Remember that if you have f (x), then f (–x) is the reflection of f (x) over the y-axis. Ex 2) Graph We can write this as f (x) = (2 –1 ) x = 2 –x It is the graph in Ex 1a) flipped over the y-axis *Note: Any function of the form f (x) = c x is one-to-one *We can reflect, dilate, & translate exponential functions – just like all our other functions. The natural exponential function is f (x) = e x e is defined as the number that approaches as n approaches infinity e ≈ ….

Ex 3) The graph of f (x) = e x is “Two Truths & a Lie” Thinking about transformations, which of these is false? Why? a) h(x) = e –x b) j(x) = –e x c) k(x) = –e –x How about this set – which is false? Why? a) p(x) = e x + 2b) q(x) = e x+2 c) r(x) = e x–2

Exponential Equations *Hints: write both expressions with the same base or factor Ex 4) Solve a) 4 3x+1 = (3x+1) = 2 5 change to base 2: 2(3x+1) = 5 6x + 2 = 5 6x = 3 x = ½ exponents: b) x 2 e 3x = 4e 3x x 2 e 3x – 4e 3x = 0 e 3x (x 2 – 4) = 0 e 3x = 0 x 2 – 4 = 0 (can’t be 0) x 2 = 4 x = ±2

Populations: Many populations are modeled using exponential equations. In general, P = P 0 (1 + r) t Ex 5) The US Census says the population in 1960 was 179,000,000 and was 203,000,000 in Determine a model & estimate the population in To find r we need the increase: 1970 pop. – 1960 pop. = 203,000,000 – 179,000,000 = 24,000,000 This is per 10 years, so per year: so P = 179,000,000( ) t In 1975, P = 179,000,000( ) 15 ≈ 217,000,000

Homework #902 Pg 450 #1, 3, 5, 7, 11, 15, 17–24, 30–34, 38, 41, 46–48