Math Problem Statement

Use the graph of the function y = f(x) to answer the questions: (a) Is f(-2) negative? (b) For which value(s) of x is f(x) = 0? (c) For which value(s) of x is f(x) ≤ 0? Write your answer using interval notation.

Solution

The image contains a graph of a function y=f(x)y = f(x) and a series of questions about the function's properties.

Let's analyze each part step by step:

  1. Question (a):
    Is f(2)f(-2) negative?
    To answer this, we need to locate x=2x = -2 on the graph and see if the function value is below the x-axis. From the graph, f(2)f(-2) appears to be negative because the graph is below the x-axis at x=2x = -2.

    Answer: Yes, f(2)f(-2) is negative.

  2. Question (b):
    For which value(s) of xx, is f(x)=0f(x) = 0?
    To find where f(x)=0f(x) = 0, we need to check the points where the graph intersects the x-axis. From the graph, the function appears to cross the x-axis at three points: x=3x = -3, x=0x = 0, and x=2x = 2.

    Answer: x=3,0,2x = -3, 0, 2.

  3. Question (c):
    For which value(s) of xx, is f(x)0f(x) \leq 0?
    This asks for the intervals where the function is either on or below the x-axis. From the graph, the function is below or equal to zero from x=3x = -3 to x=0x = 0 and from x=2x = 2 onwards. Using interval notation:

    Answer: x[3,0][2,)x \in [-3, 0] \cup [2, \infty).

Would you like further details or explanations on any part?
Here are five related questions you might find helpful:

  1. What is the significance of f(x)=0f(x) = 0 in real-world applications?
  2. How can you determine if a function is increasing or decreasing from a graph?
  3. What does it mean for a function to have a negative value at a particular xx-value?
  4. How do you express intervals where a function is negative using interval notation?
  5. How does the graph of a function help in identifying its domain and range?

Tip:
When analyzing a function graph, always check for x-axis intersections (roots), where the function is above or below the x-axis, and how it behaves at its critical points (maximums and minimums).

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Math Problem Analysis

Mathematical Concepts

Graphing
Function Analysis
Roots of Equations
Interval Notation

Formulas

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Theorems

Properties of Functions
Roots of Functions

Suitable Grade Level

Grades 9-12