Question

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

Answer 1

Answer:

Composition:   $f(x)\,^o\,h(x)=4\,\sqrt{x+4} -1$

Domain = { $x\,/\,x\geq \,-4$  }

Step-by-step explanation:

Given

$f(x)=4\,x-1\,\,\, and \,\,\, h(x)=\sqrt{x+4}$

The requested composition becomes:

$f(x)\,^o\,h(x)=f(h(x))\\f(x)\,^o\,h(x)=f(\sqrt{x+4} )\\f(x)\,^o\,h(x)=4\,\sqrt{x+4} -1$

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 $x\geq \,-4$