Given:
[tex]\begin{gathered} f(x)=\sqrt[]{x} \\ g(x)=4x+7 \end{gathered}[/tex]
To find:
[tex]g\circ f[/tex]
And its domain, follow the steps below.
Step 01: Substitute x in g(x) by f(x).
[tex]g\circ f=g(f(x))[/tex]
Then, given g(x), substitute x in this equation by f(x).
[tex]\begin{gathered} g(x)=4x+7 \\ g(f(x))=4\cdot\sqrt[]{x}+7 \\ g(f(x))=4\sqrt[]{x}+7 \end{gathered}[/tex]
Thus,
[tex]g\circ f=g(f(x))=4\sqrt[]{x}+7[/tex]
Step 02: Find the domain.
The domain is the set of all possible input values, that is, the set of all possible x-values.
Since there is no square root of negative roots, x can not be negative.
Then, the domain is x ≥ 0.
Domain:
[tex]D=\mleft\lbrace x\mright|x\ge0\}[/tex]
Answer:
gof:
[tex]g\circ f=4\sqrt[]{x}+7[/tex]
Domain:
[tex]D=\lbrace x|x\ge0\}[/tex]
