SOLUTION: I have a question that states: find the inverse of the function f(x)=3x^2+12x+6

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Question 87830: I have a question that states: find the inverse of the function f(x)=3x^2+12x+6
Answer by tutor_paul(519) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=3x%5E2%2B12x%2B6
Rename f(x) as y:
y=3x%5E2%2B12x%2B6
Solve for x:
3x%5E2%2B12x%2B6-y=0
Note that you can use the quadratic formula, with the 'c' term = 6-y:
x=%28-12%2B-sqrt%28144-%284%2A3%2A%286-y%29%29%29%29%2F6
x=%28-12%2B-sqrt%28144-%2812%2A%286-y%29%29%29%29%2F6
x=%28-12%2B-sqrt%28144-%2872-12y%29%29%29%2F6
x=%28-12%2B-sqrt%2872%2B12y%29%29%2F6
So now that I have simplified this, I see that I have a +/- in my result.
This immediately tells me that any given 'x' corresponds to 2 different y's.
Hence the inverse of this IS NOT A FUNCTION. So the answer is that
f%28x%29=3x%5E2%2B12x%2B6 has no inverse.
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Another way to do this is to apply the "Horizontal Line Test" to determine if
a function has an inverse. This test states: If any horizontal line intersects the graph of f more than once, then f does not have an inverse.
Here is the graph of your function - you can see that this does not pass the Horizontal Line Test.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B12x%2B6+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%2812%29%5E2-4%2A3%2A6=72.

Discriminant d=72 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-12%2B-sqrt%28+72+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2812%29%2Bsqrt%28+72+%29%29%2F2%5C3+=+-0.585786437626905
x%5B2%5D+=+%28-%2812%29-sqrt%28+72+%29%29%2F2%5C3+=+-3.41421356237309

Quadratic expression 3x%5E2%2B12x%2B6 can be factored:
3x%5E2%2B12x%2B6+=+3%28x--0.585786437626905%29%2A%28x--3.41421356237309%29
Again, the answer is: -0.585786437626905, -3.41421356237309. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B12%2Ax%2B6+%29




Good Luck,
tutor_paul@yahoo.com