Math Problem Statement
Solution
Let's solve each part of the problem, step by step.
(a)
For a square root function, the expression inside the square root must be non-negative, as the square root of a negative number is not defined in the set of real numbers. So, we need to solve:
Solving this inequality:
Thus, the domain of is .
(b)
For a rational function, the denominator cannot be zero, so we need to find when the denominator equals zero and exclude those values from the domain. Set the denominator equal to zero:
We need to solve this quadratic equation. Using the quadratic formula:
This gives two solutions:
So, the denominator is zero at and . We must exclude these values from the domain.
Thus, the domain is:
Summary of Domains:
- (a) Domain:
- (b) Domain:
Would you like further details or explanations on any part?
Here are 5 related questions to further explore:
- How do you solve inequalities involving square roots?
- Can we have domains that are restricted to integers?
- How do you factor quadratic expressions efficiently?
- What is the significance of domain and range in functions?
- How can we visualize the domain and range on a graph?
Tip: Always check for restrictions in both the numerator (if any square roots or logarithms) and the denominator of a function to determine the domain.
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Math Problem Analysis
Mathematical Concepts
Algebra
Square Roots
Rational Functions
Quadratic Equations
Formulas
√(a) is defined when a ≥ 0
Quadratic equation: ax^2 + bx + c = 0
Theorems
Quadratic Formula
Suitable Grade Level
Grades 9-11
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