Math Problem Statement
What is the range of f(x)=x 2 for all real x ?
Question 2Answer A. (−∞,0]
B. [0,1]
C. (−∞,∞)
D. [0,∞)
Solution
To determine the range of the function for all real :
Step-by-step Solution:
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Definition of :
- This is a quadratic function that outputs the square of any real number .
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Behavior of :
- For any real , .
- Negative values for are not possible because squaring a number always results in a non-negative value.
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Minimum value:
- The minimum value of is , which occurs when .
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Maximum value:
- There is no maximum value; as or , .
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Range:
- The set of all possible values of is , which includes and all positive numbers.
Final Answer:
D. [0, ∞)
Would you like further explanation or visualizations?
Here are some related questions for practice:
- What is the domain of ?
- Find the range of .
- How does the range of change if the domain is restricted to ?
- What is the range of ?
- Determine the range of .
Tip: To find the range of a function, analyze its behavior as , and its critical points.
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Math Problem Analysis
Mathematical Concepts
Algebra
Functions
Quadratic Functions
Range of a Function
Formulas
f(x) = x^2
Theorems
Properties of Quadratic Functions
Range Analysis for Functions
Suitable Grade Level
Grades 9-10