how to find constraints of domain
To find constraints of a domain, you need to consider the limitations or conditions that determine the valid values for the input variable(s) in a given function or equation. Here are the steps to find the constraints of a domain:
1. Identify the function or equation: Start by understanding which function or equation you are working with.
2. Determine the variables: Identify the variable(s) within the function or equation. These variables represent the input value(s) of the domain.
3. Analyze the equation or function: Examine the equation or function and look for any restrictions or conditions that limit the valid values for the variable(s). Common constraints include division by zero, negative square roots, or negative logarithm arguments.
4. Identify mathematical rules or properties: Use mathematical rules or properties to identify any specific constraints. For example, if you have an expression with a logarithm, remember that logarithms require positive arguments.
5. Solve for constraints: Solve any mathematical conditions or inequalities that arise from the equation to find the valid values. This may involve solving for zero, solving inequalities, or considering special cases.
6. Express the constraints: Once you have identified the constraints, express them using appropriate mathematical notation or plain language to indicate the range of valid values for the variable(s) in the domain.
Remember that the constraints of a domain may vary depending on the specific function or equation you are working with, so it is essential to carefully analyze the function or equation to determine the limitations on the domain.
The graph below shows a company's profit f(x), in dollars, depending on the price of pens x, in dollars, sold by the company:
CJ
150
120
90
-3 -2
Part C: Describe the constraints of the domain. (3 points)
To describe the constraints of the domain based on the given graph, we need to analyze the values on the x-axis (price of pens) and determine any limitations or restrictions.
Looking at the graph, we can observe the following constraints on the domain:
1. Minimum price constraint: The graph starts at a non-negative x-value (price) of $0. This indicates that the company does not sell pens for free or at a negative price. Therefore, the domain of the function is restricted to x ≥ 0.
2. Price intervals constraint: The x-axis values on the graph are given as discrete values ($150, $120, $90, $-3, $-2). This implies that the company only sells pens at specific price points represented by these values. Thus, the domain is limited to the specific price points mentioned on the graph ($150, $120, $90, $-3, $-2), with no other values between these points being valid inputs.
To summarize, the constraints of the domain based on the given graph are:
- Price cannot be negative: x ≥ 0.
- Price can only be specific values mentioned on the graph: x ∈ { $150, $120, $90, $-3, $-2 }.
These constraints define the valid range of x (price of pens) for the given company's profit function.
To find the constraints of a domain, follow these steps:
1. Determine the variable(s) in the domain. This will depend on the specific problem or equation you are working with.
2. Identify any restrictions or limitations on the variable(s). This can come from several sources, such as the problem statement, physical constraints, or mathematical properties.
3. Determine any restrictions on the values the variable(s) can take. This can include inequalities, exclusions, or specific ranges.
4. Use the restrictions to define the domain. Write down the constraints or inequalities that must be satisfied by the variable(s) in order to be within the domain.
5. Solve the constraints to find the valid range for the variable(s) in the domain.
Note: The steps above are general guidelines, and the specific process may vary depending on the problem or equation being considered.