Math Problem Statement
What is the function? (Referencing a graph that shows a piecewise linear function with two distinct slopes forming a 'V' shape)
Solution
The image contains a question that requires analyzing a graph of a piecewise function. Here's what I can see from the image:
- The function appears to be a piecewise linear function, with the graph showing two distinct lines connected at a vertex, forming a "V" shape.
- The question asks, "What is the function?"
- The graph appears to have two slopes:
- One negative slope on the left side.
- One positive slope on the right side.
The general form of a piecewise function can be written as:
\begin{cases} \text{slope}_1 \cdot x + \text{intercept}_1 & \text{for } x < \text{vertex} \\ \text{slope}_2 \cdot x + \text{intercept}_2 & \text{for } x \geq \text{vertex} \end{cases}$$ ### Analyzing the graph: - The vertex seems to be at $$(0, 0)$$, which means the piecewise function changes behavior at $$x = 0$$. - The line on the left (for $$x < 0$$) has a negative slope, likely $$-1$$, and passes through the origin. - The line on the right (for $$x \geq 0$$) has a positive slope, likely $$1$$, and also passes through the origin. Thus, the function can be represented as: $$f(x) = \begin{cases} -1 \cdot x & \text{for } x < 0 \\ 1 \cdot x & \text{for } x \geq 0 \end{cases}$$ This is the piecewise definition of the absolute value function $$f(x) = |x|$$, but split into two linear expressions. Would you like to explore this function in more detail, or do you have further questions? ### Related Questions: 1. What are the domain and range of the piecewise function? 2. How do you find the vertex of a piecewise function graphically? 3. Can you identify the slope from a graph without any points given? 4. How can you write a general form of a piecewise function? 5. How does the graph of an absolute value function differ from other linear graphs? **Tip:** Always identify key points on a graph, such as vertices and intercepts, to better understand the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing Functions
Formulas
f(x) = \begin{cases} -1 \cdot x & \text{for } x < 0 \\ 1 \cdot x & \text{for } x \geq 0 \end{cases}
Theorems
Definition of Piecewise Functions
Slope-Intercept Form
Suitable Grade Level
Grades 9-10