SPRING 2014 MATH 131 503 QUIZ 3 SOLUTION Problem 1 If f x x5 x3 x then find the value of f 1 3 and f f 1 2 Solution To find f 1 3 we first need to find a such that f a 3 by inspection f 1 3 hence f 1 3 1 Since the given function is a polynomial the domain of f is all real numbers 2 is in the domain of f x hence f f 1 2 2 Problem 2 Find the domain of f and f 1 given that f x ln 2 ln x Solution First let us find the domain of f since the given function is ln we want 2 ln x 0 Let us solve the above inequality for x to get the domain of f lne x 2 convert the above equation to exponential form x e 2 the domain of f is e 2 Next to find a formula for f 1 we will apply two step procedure y ln 2 ln x interchange x and y x ln 2 ln y solve for y in terms of x lne 2 lne y x 2 lne y ex lne y ex 2 y ee x 2 f 1 x ee x 2 the domain of f 1 or the range of f is all real numbers R Problem 3 Solve the equation for x e2x 3ex 2 0 Solution First we will rewrite the given equation as ex 2 3ex 2 0 continued Let us denote ex a in the above equation a 2 3a 2 0 which is a Quadratic equation to solve this we will factorize first a 2 a 1 0 the solutions are a 1 and a 2 but we know that a ex ex a ex 1 taking natural log on both sides ln ex ln 1 x ln e 0 x 0 Similarly for a 2 we get x ln 2 Problem 4 Evaluate the limit if it exists lim t 0 1 1 2 t t t Solution lim t 0 1 1 2 t t t 1 1 lim t 0 t t 1 t 1 t 1 lim t 0 t t 1 t 1 t t 1 1 lim t 0 t t 1 t lim t 0 t t 1 1 lim t 0 t 1 1 0 1 1
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