Graphs of Functions with fractional powers x^(p/q)

In summary, if the domain of the function is the real numbers, changing the value of p/q will not affect the graph's domain or range. If the domain of the function is the complex numbers, then changing the value of p/q will affect the graph's domain and range.
  • #1
confusedatmath
14
0
Can someone explain the following:

How does changing the value of p/q affect the drawing of the graph (so domain/range/shape etc)
What makes this graph an odd function?
How to work out asymptotes?

Heres a picture so you know what I'm referring to:

View attachment 1825

And below is a question dealing with this type of function. Can someone please refer to the explanations on how to go about solving these questions.

View attachment 1824
 

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  • #2
confusedatmath said:
Can someone explain the following:

How does changing the value of p/q affect the drawing of the graph (so domain/range/shape etc)
What makes this graph an odd function?
How to work out asymptotes?

Heres a picture so you know what I'm referring to:

View attachment 1825

And below is a question dealing with this type of function. Can someone please refer to the explanations on how to go about solving these questions.

View attachment 1824

Before to answering to Your question an important detail has to be specified: the cases You have proposed refer to an y=f(x) that is a real functions of real variable. If x, y or both are complex variables, then the answers are fully different...

Starting from the domain, what is the domain of an f(x)?... it is the set of real values of x for which y can effectively be computed...

Kind regards

$\chi$ $\sigma$
 
  • #3
This is pretty straightforward, isn't it? You are told to use the graphs above as models (and they are clearly assuming real numbers). Looking at them you should see that the first and second have all real numbers as domain so the third must be the correct model: [tex]x^{\frac{3}{2}}[/tex] which is the same as [tex]\sqrt{x^3}[/tex].

So the answer is either (A) or (B). But the third graph has x= 0 as its left boundary, not x= a. That tells us that we must replace "x with some simple function of x.

Looking at (A) and (B) we see that must be either x- a or x+ a. Since x= a must correspond to "x= 0" in the model, which of those, x- a or x+ a, is 0 when x= a?
 

Related to Graphs of Functions with fractional powers x^(p/q)

1. What are fractional powers in a graph of a function?

Fractional powers in a graph of a function are expressions where the variable, x, is raised to a fractional exponent, such as x^(3/4). This means that the function will have values for both positive and negative x, as well as values between integers.

2. How do fractional powers affect the shape of a graph?

The shape of a graph with fractional powers will depend on the value of the exponent, p/q. If the fraction is greater than 1, the graph will have a steeper curve compared to a linear function. If the fraction is less than 1, the graph will have a flatter curve.

3. Can fractional powers be negative?

Yes, fractional powers can be negative. This means that the function will have values for negative x as well. The negative exponent will result in the reciprocal of the function, which will be reflected across the x-axis.

4. How do you graph a function with fractional powers?

To graph a function with fractional powers, you can plot points on a coordinate plane using a table of values. Alternatively, you can use a graphing calculator or online graphing tool to plot the function. It may also be helpful to simplify the fraction before graphing to better understand the shape of the graph.

5. What are some real-life applications of functions with fractional powers?

Functions with fractional powers are often used to model real-life phenomena, such as population growth or radioactive decay. They can also be used to describe the relationship between variables in scientific experiments, such as Boyle's Law which relates the pressure and volume of a gas. Additionally, fractional powers are used in financial mathematics to calculate compound interest.

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