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Math 110 Section 4.3 An exponential function is a function f, such that where a is an actual number Page 1 / 3 f ( x) a , a 0, a 1 f ( x) 10 x f ( x) 2 x Examples: 1 f ( x) 2 Observations: Domain = Range = x x , 0, 3 f ( x) 4 x (you can choose any value you want for x) (since a must be positive, the output must be positive, too) If you graph the function, the graph will go through 1 1, a f ( x) a x f (1) a 1 0,1 f ( x) a x f (0) a 0 f ( 0) 1 1, a f ( x) a x f (1) a1 f (1) a f ( 1) 1 a For a > 1 The graph is nearly horizontal as you go left (negative) Since you're making a smaller and smaller fraction But you never actually hit the y axis It's increasingly steep as you go rightward Since you're making a larger and larger number For 0 < a < 1 The graph is nearly horizontal as you go right Since you're making a smaller and smaller fraction But you never actually hit the y axis It's increasingly steep as you go leftward Since you're making a larger and larger number Notice that 1 f ( x) 2 Math 110 x looks just like f ( x) 2 x , except reflected across the y axis Page 1 / 3 Math 110 Section 4.3 Page 2 / 3 <Draw a couple> ICE: Have them draw a couple Math 110 Page 2 / 3 Math 110 Section 4.3 Page 3 / 3 Transformations Negatives can be in two places: in the exponent, and in front of the base: 2x x in front of the base 2 in the exponent: Effects: x In the exponent: 2 This actually reflects it across the y axis In front of the base 2 Reflects is across the x axis, since we’re taking the normal output, and making it negative x Do demo: 2 x 3 1 Properties Useful for Solving Exponential / Logarithmic Equations 1. If x y x y a a If then ax ay then x y Example: 5 x 125 5 x 5 3 x = 3 5 x 1 125 5 x 1 5 3 x -1 = 3 x = 4 etc Math 110 Page 3 / 3