Math Problem Statement

Which of the following piecewise functions is graphed above?

Solution

Let's analyze the graph and the piecewise functions to determine which one matches.

Observations from the graph:

  1. For x2x \leq -2: The function appears to be linear, with a downward slope, indicating that the function could be of the form f(x)=2x3f(x) = -2x - 3.
  2. For 2<x<2-2 < x < 2: The function shows a parabolic behavior, opening upwards. This suggests the function is quadratic, likely of the form f(x)=x2f(x) = x^2 or similar.
  3. For x2x \geq 2: The graph becomes constant at y=2y = 2, indicating the function is constant for this interval.

Analyze each option:

Option A:

\begin{cases} -2x - 3 & x \leq -2 \\ x^2 + 2 & -2 < x < 2 \\ 2^x & x \geq 2 \end{cases}$$ - The first part matches the linear behavior for $$x \leq -2$$. - The second part $$x^2 + 2$$ seems incorrect because the graph intersects the y-axis at 0, not 2. - The third part $$2^x$$ is an exponential function, but the graph is constant at $$y = 2$$, so this does not match. #### Option B: $$f(x) = \begin{cases} -2x - 3 & x < -2 \\ x^2 & -2 \leq x \leq 2 \\ 2 & x > 2 \end{cases}$$ - The first part matches the linear segment for $$x < -2$$. - The second part $$x^2$$ matches the parabolic shape between $$-2 \leq x \leq 2$$. - The third part $$2$$ matches the constant part for $$x > 2$$. #### Option C: $$f(x) = \begin{cases} -2x + 3 & x < -2 \\ x^2 - 2 & -2 \leq x \leq 2 \\ 2 & x > 2 \end{cases}$$ - The first part $$-2x + 3$$ does not match the linear behavior of the graph. - The second part $$x^2 - 2$$ does not match because the graph reaches 0 at $$x = 0$$, not $$-2$$. - The third part $$2$$ is correct, but overall, this option does not match. #### Option D: $$f(x) = \begin{cases} -2x - 3 & x < -2 \\ -x^2 + 2 & -2 \leq x \leq 2 \\ 2^x & x > 2 \end{cases}$$ - The first part $$-2x - 3$$ is correct. - The second part $$-x^2 + 2$$ does not match the parabolic curve because the parabola opens upwards, not downwards. - The third part $$2^x$$ does not match the constant $$y = 2$$ in the graph. ### Conclusion: The correct piecewise function that matches the graph is **Option B**. Would you like any further details or explanations? ### Related Questions: 1. How do you determine the linear function from a graph? 2. What are the properties of quadratic functions and how can you recognize them on a graph? 3. How can you identify when a function is constant on a graph? 4. How do exponential functions compare to constant functions visually? 5. What are the general steps to analyzing a piecewise function from a graph? **Tip:** When matching a piecewise function to a graph, break down the graph into different intervals and focus on recognizing the patterns (linear, quadratic, constant, etc.) within each segment.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Quadratic Functions
Constant Functions

Formulas

f(x) = ax + b (linear equation for x ≤ -2)
f(x) = x^2 (quadratic equation for -2 ≤ x ≤ 2)
f(x) = c (constant for x > 2)

Theorems

Graphical representation of piecewise functions
Matching a graph with its corresponding algebraic function

Suitable Grade Level

Grades 9-12