Math Problem Statement

The user submitted an image containing four problems related to finding the domain and range of given functions, including graph-based and equation-based questions.

Solution

The image contains four problems related to determining the domain and range of functions, including two graph-based and two equation-based questions. I'll guide you through solving each one:

1. Domain and Range of the Graphs

Graph 1:

  • Domain (D): The set of all x-values for which the function is defined.

    • Based on the graph, the x-values seem to span from approximately x=2x = -2 to x=2x = 2 (closed interval).
    • Domain: [2,2][-2, 2]
  • Range (R): The set of all y-values that the function takes.

    • The graph's y-values range from y=0y = 0 to y=3y = 3 (closed interval).
    • Range: [0,3][0, 3]

Graph 2:

  • Domain (D): The x-values span from approximately x=4x = -4 to x=4x = 4.

    • Domain: [4,4][-4, 4]
  • Range (R): The y-values range from y=3y = -3 to y=2y = 2.

    • Range: [3,2][-3, 2]

2. Equations

Equation 1: y=3x2+1y = -3x^2 + 1

  • Domain: Since this is a quadratic function, xx can take any real number.

    • Domain: (,)(-\infty, \infty)
  • Range: As x2x^2 always gives non-negative values, the function has a maximum value at x=0x = 0, where y=1y = 1. As xx increases in either direction, yy becomes negative due to the 3x2-3x^2 term.

    • Range: (,1](-\infty, 1]

Equation 2: y=1x+3y = \frac{1}{x+3}

  • Domain: The function is undefined when x+3=0x + 3 = 0, or x=3x = -3, so all real numbers except x=3x = -3.

    • Domain: (,3)(3,)(-\infty, -3) \cup (-3, \infty)
  • Range: Since y=1x+3y = \frac{1}{x+3}, the function takes all real values except 0 (as the numerator is constant 1).

    • Range: (,0)(0,)(-\infty, 0) \cup (0, \infty)

Would you like further details on any of these, or should I clarify a particular part? Here are five related questions:

  1. What is the graphical representation of functions involving square terms like 3x2-3x^2?
  2. How do we calculate the domain for rational functions like y=1x+3y = \frac{1}{x+3}?
  3. Why does the range of a quadratic function have a maximum or minimum value?
  4. How can vertical asymptotes help determine the range of rational functions?
  5. What changes to a function can affect its domain and range?

Tip: Always look for values that might make the denominator zero or cause square roots of negative numbers when determining domain!

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Math Problem Analysis

Mathematical Concepts

Domain and Range
Graphing Functions
Quadratic Functions
Rational Functions

Formulas

Domain: Set of all x-values
Range: Set of all y-values

Theorems

Closed and open intervals
Vertical asymptotes

Suitable Grade Level

Grades 9-12