Math Problem Statement
Solution
The problem involves analyzing the graph of and identifying the properties that apply. Let’s break it down:
Key Features of :
- Graph Shape: The graph of is a "V" shape with its vertex at . It is symmetric about the y-axis.
- Increasing/Decreasing Behavior:
- decreases on the interval .
- increases on the interval .
- Domain and Range:
- Domain: All real numbers ().
- Range: Non-negative real numbers ().
- Even/Odd Function:
- is an even function because for all .
- Linear Function:
- is not linear overall, as it is piecewise linear (two linear parts joined at the vertex).
Properties to Select:
From the provided options:
- A. The function is increasing on the interval : Incorrect. increases only on .
- B. The function is an even function: Correct. is even.
- C. The domain of is : Correct.
- D. The function is an odd function: Incorrect. is even, not odd.
- E. The range of is : Incorrect. The range is .
- F. The function is a linear function: Incorrect.
- G. None of the above properties apply: Incorrect, as some properties do apply.
Correct Selections:
- B. The function is an even function.
- C. The domain of is .
Would you like a more detailed explanation of any part?
Related Questions:
- What is the difference between even and odd functions?
- How does the range of compare to its domain?
- Can a function be both even and linear? Why or why not?
- How do transformations affect the graph of ?
- What are other examples of piecewise functions, and how do their domains and ranges work?
Tip: To confirm if a function is even or odd, substitute into . If , it's even; if , it's odd.
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Math Problem Analysis
Mathematical Concepts
Absolute Value Functions
Graph Transformations
Even and Odd Functions
Domain and Range
Formulas
f(x) = |x|
Theorems
Properties of Even and Odd Functions
Domain and Range of Functions
Suitable Grade Level
Grades 9-11
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