Math Problem Statement
Solution
The image contains four problems related to determining the domain and range of functions, including two graph-based and two equation-based questions. I'll guide you through solving each one:
1. Domain and Range of the Graphs
Graph 1:
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Domain (D): The set of all x-values for which the function is defined.
- Based on the graph, the x-values seem to span from approximately to (closed interval).
- Domain:
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Range (R): The set of all y-values that the function takes.
- The graph's y-values range from to (closed interval).
- Range:
Graph 2:
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Domain (D): The x-values span from approximately to .
- Domain:
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Range (R): The y-values range from to .
- Range:
2. Equations
Equation 1:
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Domain: Since this is a quadratic function, can take any real number.
- Domain:
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Range: As always gives non-negative values, the function has a maximum value at , where . As increases in either direction, becomes negative due to the term.
- Range:
Equation 2:
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Domain: The function is undefined when , or , so all real numbers except .
- Domain:
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Range: Since , the function takes all real values except 0 (as the numerator is constant 1).
- Range:
Would you like further details on any of these, or should I clarify a particular part? Here are five related questions:
- What is the graphical representation of functions involving square terms like ?
- How do we calculate the domain for rational functions like ?
- Why does the range of a quadratic function have a maximum or minimum value?
- How can vertical asymptotes help determine the range of rational functions?
- What changes to a function can affect its domain and range?
Tip: Always look for values that might make the denominator zero or cause square roots of negative numbers when determining domain!
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Math Problem Analysis
Mathematical Concepts
Domain and Range
Graphing Functions
Quadratic Functions
Rational Functions
Formulas
Domain: Set of all x-values
Range: Set of all y-values
Theorems
Closed and open intervals
Vertical asymptotes
Suitable Grade Level
Grades 9-12
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