Math Problem Statement
Solution
You have provided two rational functions for which we are tasked with analyzing the domain, range, and asymptotes. Let's go through each one step by step.
1.
Domain:
The domain of a rational function is all real numbers except where the denominator is zero.
- Set the denominator equal to zero: Solving for : Thus, the domain is:
Vertical Asymptote:
A vertical asymptote occurs where the denominator equals zero (which would make the function undefined). From the domain, we know this occurs at:
Horizontal Asymptote:
The horizontal asymptote is found by comparing the degrees of the numerator and denominator. Both the numerator and denominator have degree 1, so we divide the leading coefficients: Thus, the horizontal asymptote is:
Range:
Since the function has a horizontal asymptote at , the range excludes this value. The range is:
2.
Domain:
To find the domain, we set the denominator equal to zero and solve for : Using the quadratic formula: Thus, the solutions are: So the domain is all real numbers except and :
Vertical Asymptotes:
Vertical asymptotes occur where the denominator equals zero, so: These are the vertical asymptotes.
Horizontal Asymptote:
The degrees of both the numerator and denominator are 2, so we again compare the leading coefficients: Thus, the horizontal asymptote is:
Range:
Since the horizontal asymptote is , the range is: [
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Domain and Range
Asymptotes
Formulas
Domain: Set the denominator ≠ 0
Vertical Asymptote: Set the denominator = 0
Horizontal Asymptote: Compare degrees of numerator and denominator
Theorems
Quadratic formula for solving denominators
Degree-based comparison for horizontal asymptotes
Suitable Grade Level
Grades 10-12
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