Decomposing a function as a composition
Find f(x) and g(x) such that h(x)=(fog)(x)
h(x)=x^{2}-3x/square root of x+5
I tried to do it but I do not think is right. this is the answer I get: f(x)= x^{2}-3x g(x)= 1/square root of x+5
Decomposing a function as a composition
Find f(x) and g(x) such that h(x)=(fog)(x)
h(x)=x^{2}-3x/square root of x+5
I tried to do it but I do not think is right. this is the answer I get: f(x)= x^{2}-3x g(x)= 1/square root of x+5
Your answer was a quotient, not a composition. You found f and g so that h(x) = f(x)/g(x).
There are an unlimited number of choices given the way the problem is stated.
For instance, there are always these trivial choices:
Choice #1) f(x) = h(x) and g(x) = x
Choice #2) f(x) = x and g(x) = h(x)
Maybe more interesting is letting g(x) = x+5.
Then and , so
, where
Choice #3)
Now try . Note that (actually, it can't be allowed to be 0). Have .
Thus and , so
Thus
(now use to know that )
, where
Choice #4)
Hi! Thanks for answering my question. However my professor said that is not the way he wants it. He said that the answer g(x) should be g(x)=1/square root of x+5 and I only need to find f(x). I guess he only wants a quotient? Mine is wrong.Can you help me please?? So when I find (fog) I will end up with: x^2-3x/ square root of x+5 back again. Thanks!!
Review what I did for Choice #3 and Choice #4. In each case, I began with choice of g(x) (one time was x+5, the other it was sqrt(x+5)). Then I did some algebra and eventually found f(t) such that h(x) = f(g(x)). So if you repeat that process, you'll solve the problem. Repeat the process I did, only now your choice of g(x) will be:
.