Explanation Given: The function is f ( x ) = x − 1 2 x + 1 . 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 − 1 2 x + 1 is defined at all the real number except the point x = − 1 2 . Thus, the domain of the function f ( x ) = x − 1 2 x + 1 is ( − ∞ , − 1 2 ) ∪ ( − 1 2 , ∞ ) . Symmetry: The given function is f ( x ) = x − 1 2 x + 1 . Substitute x = − x in f ( x ) , f ( − x ) = ( − x ) − 1 2 ( − x ) + 1 = − x + 1 − 2 x + 1 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 − 1 2 ( 0 ) + 1 = − 1 1 = − 1 That is the function f ( x ) intercepts y -axis at the point ( 0 , − 1 ) . The function f ( x ) intercepts x -axis at f ( x ) = 0 . x − 1 2 x + 1 = 0 x − 1 = 0 x = 1 Thus, x = 1 . That is the function f ( x ) intercepts x -axis at the points ( 1 , 0 ) . Asymptote: Observe that the graph of the function f ( x ) = x − 1 2 x + 1 tends to the line x = − 1 2 as the value of x tends to − 1 2 . Thus x = − 1 2 is the vertical asymptote for the function f ( x ) = x − 1 2 x + 1 . Observe the limit, lim x → ∞ f ( x ) = lim x → ∞ x − 1 2 x + 1 = lim x → ∞ 1 − 1 x 2 + 1 x = 1 2 . Thus, as the value of x tends to infinity, the graph of the function f ( x ) = x − 1 2 x + 1 tends to the line y = 1 2 . Thus y = 1 2 is the horizontal asymptote for the function f ( x ) = x − 1 2 x + 1 . Critical point: The given function is f ( x ) = x − 1 2 x + 1 . Differentiate f ( x ) with respect to x , f ′ ( x ) = d d x ( x − 1 2 x + 1 ) = ( 2 x + 1 ) ( 1 ) − ( x − 1 ) ( 2 ) ( 2 x + 1 ) 2 = 2 x + 1 − 2 x + 2 ( 2 x + 1 ) 2 = 3 ( 2 x + 1 ) 2 Observe that the function f ′ ( x ) = 3 ( 2 x + 1 ) 2 is never zero

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ISBN: 2818440028625

Author: Lial

Publisher: ELSEVIER

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