MHB Graphs of Negative Fractional Powers x^(-p/q)

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The discussion focuses on understanding the graphs of negative fractional powers, specifically x^(-p/q). The value of p/q significantly influences the shape and behavior of the graph, affecting both the domain and range. The domain is defined by the conditions where the function is valid, often excluding values that make the denominator zero or violate square root conditions. The range can be more complex to determine but can often be estimated based on the function's behavior. Asymptotes are identified by analyzing the limits of the function as it approaches critical points, such as the zeros of the denominator.
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Can someone explain how these graphs are drawn. How does the value of p/q affect this graph? How does the domain and range change? How are the asymptotes found?

The below is an image about what I'm talking about:

View attachment 1831

Here is a question the deals with this type of graph (no idea how to solve it because I'm unfamiliar with these kind of graphs)

View attachment 1832
 

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Think about the domain as where the function is defined. For example, if you think of a fraction with a polynomial in the denominator then it it is defined for all numbers except for the zeros of this polynomial. If you have a square root then it has to be defined for only positive integers. If you have both then you have to take care of both conditions. As for the range it is a little bit tricker to work with but for the most time you can guess it.

Take the following example

$$f(x)=\frac{1}{x+a}$$

What is the domain, range and horizontal and vertical asymptotes ?
 
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