Explanation Given: The function is f ( x ) = − x + 4 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 ) = − x + 4 x + 2 . Domain: The given function f ( x ) = − x + 4 x + 2 is defined at all the real number except the point x = − 2 . Thus, the domain of the function f ( x ) = − x + 4 x + 2 is ( − ∞ , − 2 ) ∪ ( − 2 , ∞ ) . Symmetry: The given function is f ( x ) = − x + 4 x + 2 . Substitute x = − x in f ( x ) , f ( − x ) = − ( − x ) + 4 ( − x ) + 2 = x + 4 − 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 ) = 0 + 4 0 + 2 = 2 That is the function f ( x ) intercepts y -axis at the point ( 0 , 2 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . − x + 4 x + 2 = 0 − x + 4 = 0 x = 4 Thus, x = 4 . That is the function f ( x ) intercepts x -axis at the points ( 4 , 0 ) . Asymptote: Observe that the graph of the function f ( x ) = − x + 4 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 ) = − x + 4 x + 2 . Observe the limit, lim x → ∞ f ( x ) = lim x → ∞ − x + 4 x + 2 = lim x → ∞ − 1 + 4 x 1 + 2 x = − 1 . Thus, as the value of x tends to infinity, the graph of the function f ( x ) = − x + 4 x + 2 tends to the line y = − 1 . Thus y = − 1 is the horizontal asymptote for the function f ( x ) = − x + 4 x + 2 . Critical point: The given function is f ( x ) = − x + 4 x + 2 . Differentiate f ( x ) with respect to x , f ′ ( x ) = ( x + 2 ) ( − 1 ) − ( − x + 4 ) ( 1 ) ( x + 2 ) 2 = − 6 ( x + 2 ) 2 Observe that the function f ′ ( x ) = − 6 ( x + 2 ) 2 is never zero

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Author: RITCHEY

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Chapter 13.4, Problem 13E

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EBK FINITE MATHEMATICS AND CALCULUS WIT

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)=-x+4x +2 is symmetric about the origin.

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