Math Problem Statement

what function is this

what function is this

Solution

The graph shown is that of the absolute value function, represented by:

y=xy = |x|

Explanation:

  1. The absolute value function creates a "V" shape centered at the origin (0,0)(0, 0).
  2. The graph is symmetric with respect to the y-axis, which means for every positive value xx, the function returns the same result as for x-x.
  3. The two arms of the graph have slopes of 11 and 1-1, indicating that:
    • On the right side (when x0x \geq 0), the slope is 11.
    • On the left side (when x<0x < 0), the slope is 1-1.

In general, the absolute value function can be written piecewise as:

\begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ This explains the linear nature of each segment of the "V" shape. --- Do you need further details or have additional questions? Here are 5 follow-up questions for you: 1. How would the graph change if the function were $$y = |x| + 2$$? 2. What transformation occurs if the function is $$y = |x - 3|$$? 3. How would scaling affect the graph, such as $$y = 2|x|$$? 4. Can you explain the domain and range of the absolute value function? 5. What is the derivative of $$y = |x|$$ at $$x = 0$$, and why is it undefined? **Tip:** Absolute value functions are often used to model situations where only positive magnitudes are relevant, such as distances or deviations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Functions
Piecewise Functions

Formulas

y = |x|
y = x if x ≥ 0, y = -x if x < 0

Theorems

Absolute Value Properties

Suitable Grade Level

Grades 7-9