Question

For the real-valued functions, find the composition and specify its domain using interval notation.

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Answer 1

Composition:
`f(x)\,^o\,h(x)=4\,√(x+4) -1`

Domain = {
`x\,/\,x\geq \,-4` }

Explanation:

Given

`f(x)=4\,x-1\,\,\, and \,\,\, h(x)=√(x+4)`

The requested composition becomes:

`f(x)\,^o\,h(x)=f(h(x))\\f(x)\,^o\,h(x)=f(√(x+4) )\\f(x)\,^o\,h(x)=4\,√(x+4) -1`

Then the Domain consists of all Real x-values for which the function is defined. In this case, the function is defined for all x-values for which the square root is defined, that is all x values for which:

`x+4\geq \,0` (so the square root is defined in the Real number system)

This means
`x\geq \,-4`

The Domain is then all the Real x-values such that
`x\geq \,-4`