Question

Graph the functions in Exercises $37-56$ $$ y=\frac{1}{(x-1)^{2}} $$

   Graph the functions in Exercises $37-56$
$$
y=\frac{1}{(x-1)^{2}}
$$

Thomas Calculus

Thomas Calculus

George B. Thomas Jr. 14th Edition

Chapter 1, Problem 53

Instant Answer

Solved by Expert Wen Zheng

07/29/2019

Step 1

The function is undefined when the denominator is zero, which happens when $x=1$. Therefore, the domain of the function is all real numbers except $x=1$. Show more…

Show all steps

lock

Thanks for your feedback!

Graph the functions in Exercises $37-56$ $$ y=\frac{1}{(x-1)^{2}} $$

Play audio

Feedback

Powered by NumerAI

Wen Zheng and 92 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. They often exhibit features like discontinuities and asymptotes due to factors in the denominator that can become zero, requiring careful analysis of their domain and behavior.

Function Transformations

Function transformations involve modifying a parent function, like a translation, reflection, stretching, or compression, to create a new function. For example, shifting the basic function 1/x² horizontally or vertically results in changes to the graph’s position without altering its overall shape.

Vertical Asymptotes

Vertical asymptotes occur when the function’s denominator equals zero, causing the function's value to approach infinity or negative infinity. They indicate where the function is undefined and help in understanding the discontinuities in the graph.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values grow very large in magnitude, indicating the value that the function tends toward in the long run. They are essential for understanding the end behavior of rational functions.

Domain and Range

The domain of a function is the set of all possible inputs for which the function is defined, while the range is the set of all possible outputs. Analyzing the domain and range of rational functions involves identifying the values excluded by vertical asymptotes and understanding the restrictions imposed by the function's structure.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later