Math Problem Statement
Solution
To find the holes of the function:
Step 1: Factor the Denominator
Factor :
Thus, the function simplifies to:
Step 2: Identify the Hole
A hole occurs where a common factor cancels in both the numerator and denominator. Since is a common factor, it cancels out, leaving:
So, there is a hole at .
Step 3: Find the y-Coordinate of the Hole
To find the corresponding -value, substitute into the simplified function:
Final Answer
The hole is at:
Would you like a further breakdown of any step?
Here are five related questions you might find useful:
- How do you determine vertical asymptotes from a rational function?
- What is the difference between a hole and a vertical asymptote?
- How do you find the domain of a rational function?
- What happens if there are multiple removable discontinuities (holes)?
- Can a rational function have both a hole and a horizontal asymptote?
Tip: Holes occur when a factor cancels, while vertical asymptotes remain when factors do not cancel.
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Math Problem Analysis
Mathematical Concepts
Rational Functions
Factoring
Removable Discontinuities
Formulas
Factoring Quadratic Expressions
Finding Holes in Rational Functions
Theorems
Removable Discontinuity Theorem
Suitable Grade Level
Grades 10-12
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