Ben wants to buy a new car stereo and he has already saved some money. He used this inequality to represent the amount he still
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Always remember two steps while finding the domain of composite function.
1) <em>First find the domain of inside/ input function( A common mistake is to skip this point).</em>
2) <em>Find the domain of new function after performing the composition.</em>
In our case, the given functions are
,
Now,
1) (f ° g)(x)
= f[g(x)]
=f[x²-81]
(this is (f ° g)(x) function)
Now, its domain
First find the domain of input function which is x²-81, its domain is the set of all real numbers.
domain of new function after performing composition which is .
x²-81=0 ⇒ (x-9)(x+9)=0
(x-9)=0 , (x+9)=0
x=9 , x=-9 (exclude these points from the domain because anything/0 does not exist in math)
So, the Domain of Composite function (f °g)(x) is the set of all real numbers except x=9, x=-9.
Domain= (-infinity, -9)U(-9, 9)U(9, infinity)
2) (g °f)(x)
= g[f(x)]
= g[3/x]
= (3/x)² -81
= 9/x² -81
First find the domain of input function 3/x which is set of all real numbers except x=0
Domain of above function after performing composition is set of all real numbers except x=0
So, the domain of (g° f)(x) is the set of all real numbers except x=0.
Domain= (-infinity, 0)U(0, infinity)
so the 2nd equation is really the first equation in disguise.
since both equations are the same, that means if you graph them, is just one line pancaked over the other, and the solutions points is every single one on each, namely infinitely many solutions.