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

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 4.3, Problem 51E

To determine

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

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 50E

Chapter 4.3, Problem 52E

Students have asked these similar questions

Question 8 Use the graph of f to evaluate the following: 6 f(x) 5 4 3 2 1 -1 1 2 3 4 5 -1 t The average rate of change of f from 4 to 5 = Question 9 10 ☑ 4p

Question 15 ✓ 6 pts 1 Details The function shown below is f(x). We are interested in the transformed function g(x) = 3f(2x) - 1 a) Describe all the transformations g(x) has made to f(x) (shifts, stretches, etc). b) NEATLY sketch the transformed function g(x) and upload your graph as a PDF document below. You may use graph paper if you want. Be sure to label your vertical and horizontal scales so that I can tell how big your function is. 1- 0 2 3 4 -1- Choose File No file chosen Question 16 0 pts 1 Details

help

Chapter 4 Solutions

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

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

Solution Summary: The author illustrates the graph of the function h(x)=x+2 x-2 by curve-sketching guide.

Concept explainers

Videos

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

Want to see the full answer?

Chapter 4 Solutions