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

10th Edition

ISBN: 8220102020252

Author: RITCHEY

Publisher: PEARSON

See similar textbooks

Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

Videos

Question

Chapter 13, Problem 43RE

To determine

To sketch: The graph of the function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 42RE

Chapter 13, Problem 44RE

Chapter 13 Solutions

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

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.

Solution Summary: The author explains that the function f(x)=x-12x+1 is symmetric about the origin.

Concept explainers

Videos

To sketch: The graph of the function.

Want to see the full answer?

Chapter 13 Solutions