Kindra M.

asked • 10/01/16

How do you find the inverse of x^2+4x-1?

I know i substitute x for y then solve for y.

Mark M.

That is exactly what you do. I suggest you do that and then post your results. You can learn that way.
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10/01/16

Kindra M.

x=y2+4y-1

x+1=y2+4y

???
 
 
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10/01/16

Kindra M.

I figured it out. I need to complete the square which will cause me to add 4 to both sides.
x+1+4=y2+4y+4
x+5    =(x+2)(x+2)
x+5    =2(x+2)   take square root of both sides
sqrt(x+5)=x+2
sqrt(x+5)-2=y
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10/01/16

Kindra M.

That should have been (x+2)2 on the third line
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10/01/16

Mark M.

It is good when you get through these! Note that while the original equation is a function, its inverse is not (when you take the √ it should be ±.
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10/01/16

1 Expert Answer

By:

Kenneth S. answered • 10/01/16

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Consider the given function, which has a parabola as its graph (opens upwardly).
This graph FAILS the Horizontal Line test and therefore it does not have an inverse.
 
(If the graph were restricted to just one part to left or right of vertex, then there would be an inverse.)
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Kindra M.

There is an inverse. Its sqrt(x+5)-2. I just don't know how to come up with it. Also isn't that suppose to be a Vertical line test?
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10/01/16

Kenneth S.

The test of a function to determine whether it has an inverse is called the Horizontal Line Test.
Since your quadratic function had no domain restrictions, it does not have an inverse.
 
When you interchanged variables & then solve for the new y in order to find an inverse, you get two answers (cf. ± in the formula), and this means for any input you have MULTIPLE outputs, so there's another way to see that this "inverse" IS NOT A FUNCTION.
 
The Horizontal Line Test is a test for "1 to 1" behavior of the original function, i.e. every function having an inverse MUST BE STRICTLY INCREASING OR STRICTLY DECREASING (mutually exclusive options).
 
As I mentioned in my Answer, there's no inverse unless the domain is restricted, and I described what that would have to be.
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10/01/16

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