Math Problem Statement
Solution
Let's analyze the problem.
Given Function:
(a) Fill in the table:
The values of for corresponding are calculated as follows:
- For ,
- For ,
- For ,
- For ,
- For ,
- For ,
- For ,
- For ,
- For ,
- For ,
The values seem mostly correct, but for , the calculated value is approximately , not , and for , it should be instead of .
(b) Are there any values of that cannot be inputted into ?
Yes, cannot be inputted because it would result in division by zero, which is undefined. The function includes , and division by zero is not allowed.
(c) Sketch of the graph of :
- As , the function approaches a vertical asymptote since tends toward for and for .
- For large positive and negative values of , the function approaches 1, creating a horizontal asymptote at .
I can help you sketch the graph based on these observations. Would you like me to generate the graph?
Do you want any further clarification or details?
Here are five related questions for deeper understanding:
- What happens to the function as approaches 0 from the positive side?
- What is the behavior of as ?
- How do vertical and horizontal asymptotes affect the shape of a graph?
- Can you identify points where the function is increasing or decreasing?
- How would you describe the domain and range of ?
Tip: Asymptotes play a key role in understanding the behavior of rational functions like , especially at extreme values of .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rational Functions
Asymptotes
Graphing
Domain and Range
Formulas
f(x) = (x + 1) / x
f(x) = 1 + 1/x
Theorems
Division by Zero Undefined
Vertical Asymptotes
Horizontal Asymptotes
Suitable Grade Level
Grades 9-12
Related Recommendation
Graph of f(x) = 1/x: Domain, Range, and Asymptotes Explained
Graph of f(x) = 1/(1-x) with Asymptote Explanation
Analysis of the Rational Function f(x) = x^2 / (x^2 - 1) - Domain, Asymptotes, Zeros
Graph of f(x) = x + 1/(x + 3): Understanding Rational Function Graphs
Evaluating the Rational Function f(x) = (2x + 4) / (x + 1)