Explanation Given: The function is f ( x ) = x + 8 x . 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: Domain: The given function f ( x ) = x + 8 x is defined at all the real number except the point x = 0 . Thus, the domain of the function f ( x ) = x + 8 x is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . Symmetry: The given function is f ( x ) = x + 8 x . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) + 8 − x = − x − 8 x Thus, f ( − x ) = − f ( x ) . Therefore, from the definition stated above, the polynomial is symmetric about the origin. Intercept: The function f ( x ) intercepts y -axis at x = 0 . But the function f ( x ) = x + 8 x is not defined at x = 0 . That is the function f ( x ) does not intercept y -axis at any point. The function f ( x ) = x + 8 x is never zero. That is the function f ( x ) does not intercept x -axis at any point. Asymptote: Observe that the graph of the function f ( x ) = x + 8 x tends to the line x = 0 as the value of x tends to 0. Thus x = 0 is the vertical asymptote for the function f ( x ) = x + 8 x . Observe the limit lim x → ∞ 8 x = 0 . Thus, as the value of x tends to infinity, the the graph of the function f ( x ) = x + 8 x tends to the line y = x . Thus y = x is the oblique asymptote for the function f ( x ) = x + 8 x . Critical point: The given function is f ( x ) = x + 8 x . Differentiate f ( x ) with respect to x , f ′ ( x ) = 1 − 8 x 2 To obtain critical points, f ′ ( x ) = 0 , 1 − 8 x 2 = 0 8 x 2 = 1 x 2 − 8 = 0 ( x − 2 2 ) ( x + 2 2 ) = 0 Thus, x = 2 2 , and x = − 2 2 . Thus, the critical points are at x = 2 2 , and x = − 2 2 . Substitute x = 2 2 , and x = − 2 2 in f ( x ) = x + 8 x and obtain the critical points as ( 2 2 , 4 2 ) , and ( − 2 2 , − 4 2 ) . Relative Maxima and Relative Minima : The critical points are at x = 2 2 , and x = − 2 2 . Substitute x = − 3 in f ′ ( x ) = 1 − 8 x 2 , f ′ ( − 3 ) = 1 − 8 ( − 3 ) 2 = 1 − 8 9 = 1 9 > 0 Thus, the function f ( x ) is increasing on the interval ( − ∞ , − 2 2 )

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Finite Mathematics and Calculus with Applications

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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Solution Summary: The author explains that the function f(x)=x+8x is symmetric about the origin.

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