Vertical Asymptote at x= 4 and a hole when x = -3
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A vertical asymptote occurs at a point in a graph where the function approaches infinity or negative infinity as the input approaches a certain value. Similarly, a hole, or removable discontinuity, occurs when a function appears to have a point missing from the graph at a certain input, but can be algebraically simplified to fill in the missing point.
Given that the function has a vertical asymptote at x=4 and a hole when x=-3, we can make the following observations:
1. Vertical Asymptote at x=4:
This means that as x gets closer and closer to 4, the value of the function gets extremely large in the positive or negative direction, depending on the behavior of the function. We can represent this mathematically by writing:
lim f(x) = +/- infinity
x->4
2. Hole when x=-3:
This means that there appears to be a missing point on the graph at x=-3. However, we can algebraically simplify the function to fill in this point. Let’s say that the function is of the form f(x) = g(x) / h(x). If h(-3) is equal to zero, then we can factor out (x+3) from the denominator to obtain:
f(x) = g(x) / (x+3) * (h(x)/(x+3))
Now, we notice that the (x+3) terms cancel out in the numerator and denominator, leaving us with:
f(x) = g(x) / h(x)
This is a simplified form of the function that no longer has a hole at x=-3.
Overall, we can conclude that the function has a vertical asymptote at x=4 and a removable discontinuity at x=-3 that can be filled in by algebraically simplifying the function.
More Answers:
Creating a Function with a Hole at x = -1 and Vertical Asymptote at x = 1Understanding Vertical Asymptote and Holes in Mathematical Functions: An example with f(x) = (x-1)/(x-1)(x-2)
Understanding Horizontal and Vertical Asymptotes in Math: Finding a Function with Both