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Unformatted text preview: Math 113 Exam #1 Solutions 1. What are the domain and range of the function f ( x ) = e x √ x 4 1 ? Answer: Notice that √ x 4 1 is only defined when x 4 1 is nonnegative, meaning that x ≤  1 or x ≥ 1. Moreover, the function f is only defined when the denominator is nonzero, which excludes x = ± 1. Therefore, the domain of f is { x : x < 1 or x > 1 } = (∞ , 1) ∪ (1 , + ∞ ) . As for the range, notice that both the numerator and the denominator are always positive, so negative numbers and zero are definitely not in the range. As x gets very negative, the numerator gets very close to zero. Also, as x gets close to 1, the denominator gets very close to zero while the numerator goes to 1 /e > 0. Hence, the range of f consists of all positive numbers: Range( f ) = { y : 0 < y } = (0 , + ∞ ) . 2. Let f ( x ) = e 3 x 2 . Is f invertible? Why or why not? If f is invertible, what is f 1 ( x )? Answer: Yes, f is invertible. To see why, notice, first of all, that 3 x 2 is an increasing function and passes the horizontal line test. Moreover, e x is also an increasing function that passes the horizontal line test. Hence, the composition f ( x ) = e 3 x 2 is also an increasing function, so it has an inverse....
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 Fall '08
 CHESTKOFSKY
 Math, Calculus, lim, horizontal line test

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