Math Problem Statement

I have Df=[-4;4] and Wf=[-1;5]. f(2)=1 and f(-3)>0. now i have to draw a graph

Solution

To draw a graph of the function f(x)f(x) with the given conditions, we need to translate this information into mathematical insights:

Given:

  • Domain: Df=[4,4]D_f = [-4, 4]. The function f(x)f(x) is defined from x=4x = -4 to x=4x = 4.
  • Range: Wf=[1,5]W_f = [-1, 5]. The function's values (outputs) are between 1-1 and 55, inclusive.
  • Point 1: f(2)=1f(2) = 1. This gives us a specific point on the graph: (2,1)(2, 1).
  • Point 2: f(3)>0f(-3) > 0. This indicates that the value of the function at x=3x = -3 is positive, but we don't know the exact value. However, it must lie in the interval (0,5](0, 5].

Steps to draw the graph:

  1. Domain and Range: The graph should cover the x-values from 4-4 to 44 and the y-values from 1-1 to 55.
  2. Point (2, 1): Plot the point (2,1)(2, 1) on the graph.
  3. Point near (-3, positive): We know that f(3)>0f(-3) > 0, so plot a point at (3,y)(-3, y), where y>0y > 0. It must be between 0 and 5, but without exact information, we can choose a reasonable value (e.g., f(3)=2f(-3) = 2).
  4. General Shape: Since no other specific behavior of the function is given (e.g., if it's linear or quadratic), you can sketch a general curve that passes through or near these points and stays within the given range for the output. Ensure that the graph does not exceed the values of y=1y = -1 or y=5y = 5.

Would you like me to generate a visual graph based on this, or do you need further clarification?

Here are some questions to explore further:

  1. Should we assume the function is continuous or can it have breaks?
  2. Could we assume symmetry in the function (e.g., even or odd)?
  3. What happens at the boundary points x=4x = -4 and x=4x = 4?
  4. Do we know anything about the slope or concavity of the function?
  5. Can we infer any other points on the graph based on the domain and range?

Tip: When sketching graphs, always plot known points first, then infer the shape of the function by considering any restrictions on its range and behavior.

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Math Problem Analysis

Mathematical Concepts

Functions
Graphing
Domain and Range

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12