Math Problem Statement
Solution
In the image provided, the task involves finding the quotient of two functions and and then simplifying the result. The functions given are:
Part (a) - Find and simplify the result.
We begin by forming the quotient:
Now, let's simplify this expression:
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Factor the numerator :
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Factor the denominator : To factor this, find two numbers that multiply to and add to . These numbers are and :
Thus, the quotient becomes:
- Cancel out common factors in the numerator and the denominator:
The simplified result is:
Part (b) - State the domain of
The domain of the quotient function is determined by the restrictions that make the denominator zero. From the expression , the denominator equals zero at and .
Therefore, the domain excludes these two values. In set-builder notation, the domain is:
Would you like further details on any part of this solution?
5 Related Questions:
- How do you find the domain of a rational function?
- What are the steps to factor quadratic expressions?
- How do you simplify complex rational expressions?
- What happens if there are no common factors to cancel in a rational function?
- How do you verify the cancellation of factors in a rational function?
Tip: Always check for values that make the denominator zero before simplifying rational expressions, as these values are excluded from the domain.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Functions
Factoring
Formulas
f(x)/g(x) = (4x + 20) / (x^2 - 3x - 40)
Theorems
Factoring Quadratics
Suitable Grade Level
Grades 9-11
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