Math Problem Statement

Use the graph of d(t) = 100 - 40t in Exercise 7 to identify the domain and range of the function. Then tell whether the domain, the range, neither, or both are affected by the changes described in each part.

Solution

To solve this problem, let's break down the given information.

  1. Function Definition: The function given is d(t)=10040td(t) = 100 - 40t.

    Here, d(t)d(t) represents the output of the function, and tt is the input variable, which may represent time or another independent variable.

  2. Domain and Range:

    • Domain: The domain represents all possible values of tt for which d(t)d(t) is defined. Since there are no restrictions on tt in this linear function (no square roots, logarithms, etc.), the domain of d(t)d(t) is all real numbers: (,)(-\infty, \infty).
    • Range: The range is the set of all possible values of d(t)d(t). Since this is a linear function with a non-zero slope (40-40), d(t)d(t) can take on any real value depending on tt. Thus, the range is also all real numbers: (,)(-\infty, \infty).
  3. Effect of Changes:

    • The question asks to determine whether the domain, range, neither, or both are affected by any changes. Since I don’t have the exact changes from Exercise 7, I can only describe the general approach:
      • If changes affect the values tt can take (e.g., restricting tt to positive values), the domain would be affected.
      • If changes alter the outputs of d(t)d(t) (e.g., adjusting the slope or intercept of the function), then the range would likely be impacted.
      • In some cases, both domain and range could be affected (e.g., if tt is restricted to certain values that also impact the output range).

If you have details from Exercise 7 about specific changes, I can provide more precise insights on how each change affects the domain and range.

Would you like further clarification, or do you have additional details on Exercise 7?

Here are five related questions for further exploration:

  1. How do changes to the slope in a linear function affect the domain and range?
  2. What happens to the domain and range if we restrict tt to non-negative values?
  3. How can we interpret the domain and range in real-world applications of linear functions?
  4. Are there cases where a linear function has a restricted range but an unrestricted domain?
  5. How does altering the intercept of a linear function impact its range?

Tip: In linear functions like d(t)=10040td(t) = 100 - 40t, the domain is often unrestricted, but the range depends heavily on the slope and intercept.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Domain and Range
Graph Analysis

Formulas

d(t) = 100 - 40t

Theorems

Linear Function Properties

Suitable Grade Level

Grades 9-10