Understanding Vertical Asymptotes and Holes: Explanation and Equation Creation

Vertical Asymptote at x = 2, hole when x = 1

To understand vertical asymptotes and holes, let’s first clarify what they are

To understand vertical asymptotes and holes, let’s first clarify what they are.

A vertical asymptote is a vertical line that a function approaches but never touches as the input values (x) get closer and closer to a certain value. In this case, the vertical asymptote is at x = 2, which means that as x approaches 2, the function’s y-values will increase or decrease infinitely, but it will never actually reach the line x = 2.

A hole, on the other hand, is a point where the function has a “missing” value. It’s a point that is excluded from the graph because there is a discontinuity. In this case, we have a hole at x = 1, which means that the function is undefined at x = 1.

To determine the equation of a function with a vertical asymptote at x = 2 and a hole at x = 1, we can start with a basic rational function, such as:

f(x) = (x – k)/(x – j)

Where k and j are constants. We want the vertical asymptote at x = 2, so the denominator (x – j) should be zero at x = 2. Therefore, j = 2, and our equation becomes:

f(x) = (x – k)/(x – 2)

Next, we need to find the value of k that creates a hole at x = 1. For this, we know that the numerator (x – k) has to be zero at x = 1. Therefore, k = 1, and our final equation becomes:

f(x) = (x – 1)/(x – 2)

This function has a vertical asymptote at x = 2, meaning that as x approaches 2, the graph of this function will increase or decrease indefinitely. Additionally, there is a hole at x = 1, which is a point that is excluded from the graph because the function is undefined at that point.

I hope this explanation clarifies the concept of vertical asymptotes and holes and helps you understand the process of creating an equation with these characteristics. Let me know if you have any further questions!

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