Math Problem Statement
Solution
The image contains two rational functions for which you need to determine the domain, range, and asymptotes. Here are the functions:
Let's break them down one by one.
1.
-
Domain: The domain is all real numbers except where the denominator is zero. Solve : So, the domain is , .
-
Vertical Asymptote: A vertical asymptote occurs where the denominator is zero. In this case, there is a vertical asymptote at .
-
Horizontal Asymptote: Compare the degrees of the numerator and the denominator. Since both are linear (degree 1), the horizontal asymptote is determined by the ratio of the leading coefficients: Thus, the horizontal asymptote is .
-
Range: The range of the function excludes the horizontal asymptote , so the range is .
2.
-
Domain: The domain excludes values where the denominator is zero. Factor the denominator: So, the domain is all real numbers except and .
-
Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero. From the factored form, we get vertical asymptotes at:
-
Horizontal Asymptote: Since the degrees of the numerator and denominator are the same (degree 2), the horizontal asymptote is determined by the ratio of the leading coefficients: So, the horizontal asymptote is .
-
Range: The range excludes the horizontal asymptote , so the range is .
Would you like more details or explanations on any of these points?
Related Questions:
- How do you determine slant (oblique) asymptotes for rational functions?
- What is the general method to find the domain of any rational function?
- Can a rational function have both a horizontal and slant asymptote?
- What are the conditions for a rational function to have no horizontal asymptote?
- How do you determine the behavior of a function near its vertical asymptote?
Tip: Always check if a rational function can be simplified before analyzing its asymptotes and other characteristics.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rational Functions
Domain and Range
Asymptotes
Formulas
Domain: Set of all possible x-values except where the denominator is 0.
Vertical Asymptotes: Occur where the denominator equals 0.
Horizontal Asymptotes: Determined by the degrees of the numerator and denominator.
Theorems
Rational Function Theorem
Asymptote Theorem
Suitable Grade Level
Grades 10-12