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

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SUMMARY

The discussion focuses on the graphical representation of negative fractional powers, specifically the function \( f(x) = \frac{1}{x+a} \). It emphasizes the importance of understanding the effects of the fraction \( p/q \) on the graph's domain and range, as well as the identification of asymptotes. The domain is defined by the values for which the function is valid, excluding points where the denominator equals zero, while the range can be inferred based on the behavior of the function. Asymptotes are determined by analyzing the limits of the function as \( x \) approaches critical values.

PREREQUISITES
  • Understanding of rational functions and their properties
  • Knowledge of domain and range concepts in mathematics
  • Familiarity with asymptotes and their significance in graphing
  • Basic skills in interpreting mathematical expressions and functions
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  • Study the properties of rational functions in detail
  • Learn how to determine the domain and range of complex functions
  • Explore the concept of asymptotes in various types of functions
  • Practice graphing negative fractional powers and their transformations
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Students, educators, and anyone interested in advanced algebraic concepts, particularly those dealing with rational functions and their graphical representations.

confusedatmath
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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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