How can I sketch the reciprocal of a function with poles at x=-2 and x=2?

In summary, the reciprocal of the given function will have roots at -2 and 2 and will be symmetric about the y-axis. It will start at x=0 with a value slightly less than -1 and will increase as x goes to +infinity. The graph will resemble a quadratic graph and the middle portion will be similar to the graph of -sec x, with its reciprocal being similar to -cos x.
  • #1
danago
Gold Member
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Given the following graph:
http://img245.imageshack.us/img245/2395/scan0001ou4.gif [Broken]

How can i sketch the reciprocal of that function? There are poles at x=-2 and x=2, so it means its reciprocal will have roots at -2 and 2 right? But that's not really enough information to compose a full sketch.

How should i go about doing this?

Thanks,
Dan.
 
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  • #2
It's enough for a rough sketch. You can see that the function is symmetric about the y-axis so its reciprocal will be. You can see that at x= 0, the function has value just a little larger than -1 so its reciprocal will have value just a little less than -1. The reciprocal graph will start at x= 0, y= a little less than -1, rise to x= 2, y= 0, then continue increasing as x goes to + infinity. Use symmetry to get the graph for negative x.
 
  • #3
replace each y with 1/y for all x.
eg. where y is tending to infinity, it should tend to 0.
 
  • #4
So it will be similar to a quadratic graph?
 
  • #5
danago said:
So it will be similar to a quadratic graph?
the middle portion is very similar to graph of -sec x, so its reciproca would be similar to -cos x
 
  • #6
1/big = small

1/small = big
 

1. What is the definition of the reciprocal of a function?

The reciprocal of a function is a new function that is created by taking the inverse of the original function. In other words, the reciprocal of a function f(x) is denoted as 1/f(x) and it represents the value obtained by dividing 1 by the output of the original function.

2. How is the reciprocal of a function graphically represented?

The reciprocal of a function is graphically represented as a hyperbola, with the horizontal and vertical asymptotes at x = 0 and y = 0, respectively. The curve of the reciprocal function approaches these asymptotes but never touches them.

3. Can any function have a reciprocal?

No, not all functions have a reciprocal. For a function to have a reciprocal, its output must never be equal to 0. This means that the original function must not have any x-values that result in a y-value of 0.

4. How is the reciprocal of a function calculated?

The reciprocal of a function is calculated by finding the inverse of the original function and then taking the reciprocal of the output. For example, if the original function is f(x) = 2x, then its reciprocal would be 1/f(x) = 1/(2x) = 1/2x.

5. What is the relationship between a function and its reciprocal?

The reciprocal of a function and the original function are inversely proportional to each other. This means that as the output of the original function increases, the output of the reciprocal function decreases, and vice versa. Additionally, the domain and range of a function and its reciprocal are interchanged.

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