Explanation Given: The function is f ( x ) = 3 x x − 2 . Definition used: (1) If the function gets larger and larger in magnitude without bound as x approaches the number k, then the line x = k is the vertical asymptote. (2) A function f ( x ) said to be symmetric about the origin if f ( − x ) = − f ( x ) . Calculation: The given function is f ( x ) = 3 x x − 2 . Domain: The given function f ( x ) = 3 x x − 2 is defined at all the real number except the point x = 2 . Thus, the domain of the function f ( x ) = 3 x x − 2 is ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . Symmetry: The given function is f ( x ) = 3 x x − 2 . Substitute x = − x in f ( x ) , f ( − x ) = 3 ( − x ) − x − 2 = 3 x x + 2 Thus, f ( − x ) ≠ − f ( x ) . Therefore, from the definition stated above, the function does not have symmetry about the origin. Intercept: The function f ( x ) intercepts y -axis at x = 0 . Substitute the value x = 0 in f ( x ) , f ( 0 ) = 3 × ( 0 ) 0 − 2 = 0 That is the function f ( x ) intercepts y -axis at the point ( 0 , 0 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . 3 x x − 2 = 0 3 x = 0 x = 0 Thus, x = 0 . That is the function f ( x ) intercepts x -axis at the points ( 0 , 0 ) . Asymptote: Observe that the graph of the function f ( x ) = 3 x x − 2 tends to the line x = 2 as the value of x tends to 2. Thus x = 2 is the vertical asymptote for the function f ( x ) = 3 x x − 2 . Observe the limit, lim x → ∞ f ( x ) = lim x → ∞ 3 x x − 2 = lim x → ∞ 3 1 − 2 x = 3 . Thus, as the value of x tends to infinity, the graph of the function f ( x ) = 3 x x − 2 tends to the line y = 3 . Thus y = 3 is the horizontal asymptote for the function f ( x ) = 3 x x − 2 . Critical point: The given function is f ( x ) = 3 x x − 2 . Differentiate f ( x ) with respect to x , f ′ ( x ) = ( x − 2 ) ( 3 ) − ( 3 x ) ( 1 ) ( x − 2 ) 2 = − 6 ( x − 2 ) 2 Observe that the function f ′ ( x ) = − 6 ( x − 2 ) 2 is never zero

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 13.4, Problem 14E

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To sketch: The graph of the function.

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Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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The graph of the function.

Solution Summary: The function f(x) is symmetric about the origin if it gets larger in magnitude without bound as x approaches the number k.

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Chapter 13 Solutions