Math Problem Statement
Solution
The function provided in the image is:
1. Domain:
The domain of a function refers to all the possible input values (in this case, and ) for which the function is defined.
For a square root function, the expression inside the square root must be non-negative. Therefore, we need:
Solving this inequality:
Thus, the domain of the function is all pairs where .
2. Range:
The range of a function is the set of all possible output values.
Since the function is a square root, the output of the square root is always non-negative. Therefore, the range of the function is:
Thus, the range of the function is all non-negative real numbers .
3. Sketching the Domain:
To sketch the domain, you would plot the region in the -plane. This represents the area below or on the parabola .
- The curve is a parabola that opens upwards.
- The domain includes all points below or on this curve.
Do you need further explanation or help sketching the domain?
Related Questions:
- How do you determine the domain of other types of functions (like logarithmic or rational functions)?
- What happens if the expression inside the square root was instead of ?
- Can a function have an empty domain? What would that mean?
- How do you determine the range for functions other than square roots?
- How do transformations (like scaling or translating) affect the domain and range of functions?
Tip:
Always check for restrictions in a function (such as square roots or divisions) when determining the domain!
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Math Problem Analysis
Mathematical Concepts
Functions
Inequalities
Domain and Range
Square Root Function
Formulas
x^2 - 2y >= 0
y <= x^2 / 2
Theorems
The Square Root Function Theorem
Domain and Range Determination
Suitable Grade Level
Grade 11-12