For the given functions f, g and h find fogoh and state the exact domain of fogoh f(x)=1/x g(x)=lnx h(x)=2x+15. Need correct answer.
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Given the following functions:
f(x)= 1/x
g(x)= ln x
h(x)= 2x+15
We need to find the composite function: fogoh(x)
==> fogoh(x)= f(g(h(x)))
= f(g(2x+15))
= f(ln(2x+15))
= 1/ln(2x+15).
==> fogoh(x)= 1/ln(2x+15)
Now we will find the domain.
==> The domain is all real numbers except when the denominator is 0.
But we know that "ln" is always a positive number.
Also, 2x+15 must be greater than 0.
==> 2x+15>0
==> 2x >-15
==> x >-15/2
`==gt x in (-15/2, oo)` .............(1)
Now we need to include the domain of the input functions. h(x) and g(x).
==> The domain of g(x) is all real numbers greater than 0.
`==gt x in (0,oo)` .............(2)
==> The domain of h(x) is all real numbers
`==gt x in (-oo,oo).` .........(3)
Then we will find the common domain from (1) , (2) , and (3).
`==gt x in (-oo,oo)nn(0,oo)nn(-15/2, oo)`
`==gt x in (0,oo)`
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