Question 87830
{{{f(x)=3x^2+12x+6}}}
Rename f(x) as y:
{{{y=3x^2+12x+6}}}
Solve for x:
{{{3x^2+12x+6-y=0}}}
Note that you can use the quadratic formula, with the 'c' term = 6-y:
{{{x=(-12+-sqrt(144-(4*3*(6-y))))/6}}}
{{{x=(-12+-sqrt(144-(12*(6-y))))/6}}}
{{{x=(-12+-sqrt(144-(72-12y)))/6}}}
{{{x=(-12+-sqrt(72+12y))/6}}}

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(x)=3x^2+12x+6}}} has no inverse.

-----------------------------------------------------------------

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.

 *[invoke quadratic "x", 3, 12, 6 ]



Good Luck,
tutor_paul@yahoo.com