Math Problem Statement
Y=(x^2-4)/(2-x)
Solution
To analyze the function , let's break down the expression and explore its properties.
Step 1: Simplify the numerator
First, notice that the numerator can be factored using the difference of squares:
So, we can rewrite the function as:
Step 2: Simplify the expression
Next, observe that can be written as :
Substitute this into the expression:
Cancel the common factor in the numerator and the denominator, remembering to account for the negative sign:
However, the cancellation of implies that . So the function simplifies to:
Step 3: Consider the domain
The function is not defined when , which occurs when .
Thus, the simplified function is valid for all except .
Summary
The function simplifies to:
with the restriction:
Would you like further details or have any questions?
Here are 8 related questions to expand your understanding:
- How do you factorize the difference of squares?
- Why is it important to consider the domain when simplifying rational expressions?
- Can you find the domain and range of the simplified function ?
- What happens to the graph of a function at points where the denominator is zero?
- How do you handle simplifying expressions with common factors in the numerator and denominator?
- Can you explain how to identify vertical asymptotes in rational functions?
- How would the function change if the numerator were instead of ?
- Can you graph the function and identify any discontinuities?
Tip: Always consider the domain of the original function before simplifying, especially identifying values that make the denominator zero to avoid undefined expressions.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Rational Expressions
Factoring
Domain Restrictions
Formulas
-
Theorems
-
Suitable Grade Level
Grades 10-12