Math Problem Statement
find the domain and range of each function 1+x^2
Solution
To find the domain and range of the function , let's analyze its properties.
Domain:
The domain of a function consists of all possible values of for which the function is defined.
- The given function is a polynomial function.
- Polynomial functions are defined for all real numbers.
Domain: or all real numbers.
Range:
The range consists of all possible values of .
- The term represents a squared function, which is always non-negative ().
- Adding 1 to shifts the graph upward by 1.
- The minimum value occurs at , where .
- As , , meaning .
Thus, the function takes all values greater than or equal to 1.
Range: .
Would you like a graph or further explanation?
Related Questions:
- What is the domain and range of ?
- How does shifting affect the range of a quadratic function?
- What are the key characteristics of a quadratic function?
- How do you determine the minimum or maximum value of a quadratic function?
- What happens to the range if we modify to ?
Tip:
For quadratic functions of the form , the range is determined by the vertex. If , the function has a minimum at the vertex. If , the function has a maximum.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Domain and Range
Formulas
f(x) = 1 + x^2
Theorems
Properties of Polynomial Functions
Range of Quadratic Functions
Suitable Grade Level
Grades 7-9
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