In mathematics, the domain of a function is the set of all possible input values, or x-values, for which the function is defined. The function may have certain restrictions or conditions that limit the values of x it can take. In this case, we have a function with the domain r, and we need to determine the values of k for which this domain is valid.

Let’s assume our function is f(x) and it is defined for the domain r. So, we can represent this as:

f(x) = r for some values of x.

Now, we need to investigate the potential limitations or conditions on x imposed by the value of k. To achieve this, we need to consider the properties of the function and analyze how k affects its domain.

Analysis of the function and the role of k

The specific function in focus is not mentioned, so let’s consider a generic function for the sake of explanation. Given a function f(x) = kx, where k is a constant, we can analyze the role of k in determining the domain.

For this function to have a domain r, there should not be any restrictions on the set of possible x-values. In other words, any real number should be applicable for x.

Since the function f(x) = kx is a linear function, it represents a straight line on a graph. The slope of this line is determined by the value of k.

Determining the valid values of k

To determine the values of k for which the function has the domain r, we need to ensure that the line defined by the function does not intersect the x-axis at any point within the given range r.

If the line does intersect the x-axis, it means that there are certain x-values for which the function is not defined, violating the condition of having a domain r. This usually happens when the slope of the line is zero (i.e., a horizontal line) or undefined (i.e., a vertical line).

For our generic function f(x) = kx, the slope of the line is defined by k. If the slope is not zero or undefined, the line will never intersect the x-axis within the domain r. Therefore, any non-zero, real value of k would satisfy the condition.

In conclusion, the values of k for which the function has the domain r are any non-zero, real numbers. As long as the slope of the line represented by the function is non-zero and defined, the function will have the domain r without any restrictions on the valid x-values. However, it’s important to note that this conclusion is derived based on the assumption of a generic linear function. When dealing with a specific function with constraints or additional conditions, a detailed analysis is required to determine the valid values of k for the domain r.

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