Explanation The given function is, h ( x ) = x + 2 x − 2 (1) The domain of the function is the set of all real numbers except the number that is undefined on the function. As 2 is the number that is undefined on the given function, thus the domain of the given function is ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . It is known that y = h ( x ) . Substitute 0 for x in the given function to calculate the y -intercept. y = 0 + 2 0 − 2 y = 2 − 2 y = − 1 Substitute 0 for y in the function to calculate the x -intercept. 0 = x + 2 x − 2 0 = x + 2 x = − 2 This shows that x -intercept is − 2 and y -intercept is − 1 . To determine the behavior of the function for large absolute values of x , take the limit of x to ∞ and − ∞ . Take the limit x approaches to ∞ . lim x → ∞ = x + 2 x − 2 = 1 + 2 x 1 − 2 x [ Take x common from numerator and denominator ] = 1 Take the limit x approaches to − ∞ . lim x → − ∞ = x + 2 x − 2 = 1 + 2 x 1 − 2 x [ Take x common from numerator and denominator ] = 1 Thus, the graph of h approaches the line y = 1 as x becomes arbitrarily large. Since the function tends to ∞ for x = 2 , thus the function has a vertical asymptote at x = 2 . Since the function has the line y = 1 for x at ∞ , thus, the horizontal asymptote is y = 1 . Differentiate equation (1) with respect to x . h ′ ( x ) = ( x − 2 ) d d x ( x + 2 ) − ( x + 2 ) d d x ( x − 2 ) ( x − 2 ) 2 h ′ ( x ) = ( x − 2 ) ( 1 ) − ( x + 2 ) ( 1 ) ( x − 2 ) 2 h ′ ( x ) = x − 2 − x − 2 ( x − 2 ) 2 h ′ ( x ) = − 4 ( x − 2 ) 2 (2) This shows that it is discontinuous at x = 2 . Thus, the intervals are ( − ∞ , 2 ) and ( 2 , ∞ )

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.3, Problem 51E

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To sketch: The graph of the function h(x)=x+2x−2 by curve-sketching guide.

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Chapter 4.3, Problem 50E

Chapter 4.3, Problem 52E

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The graph of the function h ( x ) = x + 2 x − 2 by curve-sketching guide.

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To sketch: The graph of the function h(x)=x+2x−2 by curve-sketching guide.

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