Math Problem Statement

Sketch the graph of the following function and identify all properties that apply. f(x) = |x|. Identify the properties: Select all that apply.

Solution

The problem involves analyzing the graph of f(x)=xf(x) = |x| and identifying the properties that apply. Let’s break it down:

Key Features of f(x)=xf(x) = |x|:

  1. Graph Shape: The graph of f(x)=xf(x) = |x| is a "V" shape with its vertex at (0,0)(0, 0). It is symmetric about the y-axis.
  2. Increasing/Decreasing Behavior:
    • f(x)f(x) decreases on the interval (,0](-\infty, 0].
    • f(x)f(x) increases on the interval [0,)[0, \infty).
  3. Domain and Range:
    • Domain: All real numbers ((,)(-\infty, \infty)).
    • Range: Non-negative real numbers ([0,)[0, \infty)).
  4. Even/Odd Function:
    • f(x)f(x) is an even function because f(x)=f(x)f(-x) = f(x) for all xx.
  5. Linear Function:
    • f(x)f(x) 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 (,)(-\infty, \infty): Incorrect. f(x)f(x) increases only on [0,)[0, \infty).
  • B. The function ff is an even function: Correct. f(x)=xf(x) = |x| is even.
  • C. The domain of ff is (,)(-\infty, \infty): Correct.
  • D. The function ff is an odd function: Incorrect. f(x)f(x) is even, not odd.
  • E. The range of ff is (,)(-\infty, \infty): Incorrect. The range is [0,)[0, \infty).
  • F. The function ff is a linear function: Incorrect.
  • G. None of the above properties apply: Incorrect, as some properties do apply.

Correct Selections:

  • B. The function ff is an even function.
  • C. The domain of ff is (,)(-\infty, \infty).

Would you like a more detailed explanation of any part?


Related Questions:

  1. What is the difference between even and odd functions?
  2. How does the range of f(x)=xf(x) = |x| compare to its domain?
  3. Can a function be both even and linear? Why or why not?
  4. How do transformations affect the graph of f(x)=xf(x) = |x|?
  5. 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 x-x into f(x)f(x). If f(x)=f(x)f(-x) = f(x), it's even; if f(x)=f(x)f(-x) = -f(x), 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