composite functions

Suppose you are given three functions defined by their formula:

f(x)= sqrt(x^2+1)

g(x)= sqrt(x−5)

h(x) = sqrt(4cos(x)).

What is the domain of each of these functions? Simplify the result of f ◦ g ◦ h(x) and h ◦ f ◦ g(x). What is the domain and the range in each case?

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I understand the domains of f(x) and h(x)is x is an element of all real numbers, while the domain of g(x) is x >= 5. I also know that f ◦ g ◦h(x) = 2*sqrt(cosx-1) and h ◦ f ◦ g(x) = 4cos(sqrt(x-4)). But I don't understand how I can get the domain and ranges of these composite functions without graphing.

### 1 Answer | Add Yours

`f(x)=sqrt(x^2+1)`

Domain : `RR` because `x^2+1` is always positive.

Range: `{y | y gt= 1}` because `x^2+1gt=1` and ```sqrt(a)` is always positive (principal square root.

`g(x)=sqrt(x-5)`

Domain: `{x|xgt=5}` because `sqrt(x-5)` is not a real number if `xlt=5`

Range: `{y|ygt=0}` because `sqrt(a) gt= 0` (principal square root)

`h(x)=sqrt(4cos(x))`

Domain: `{x|-pi/2+2pinlt=xlt=pi/2+2pin,n in ZZ}` The domain is only where `cos(x) gt= 0` , otherwise the argument to square root is negative.``

Range: `{y|0lt=ylt=2}` The maximum of cos(x) is 1 so the maximum of the function is `sqrt(4(1)) = 2` . The maximum of the `sqrt(a)` function is `gt= 0`

f(g(h(x)))

Domain: Empty Set. Because the Range of h(x) is numbers <= 2 and when substituted into g(x) we would get a negative argument to the ``square root function.

Range: Empty set because no allowed inputs.

Hope this helps...

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