Find gof and the domain of composite function.
$f ( x ) = x ^ { 2 } + 2 , \quad g ( x ) = \sqrt { x }$
A) $( x + 2 ) ^ { 2 }$
Domain of gof: all real numbers $x$
B) $( x - 2 ) ^ { 2 }$
Domain of $g \circ f$ : all real numbers $x$
C) $\sqrt { x ^ { 2 } + 2 }$
Domain of $g \circ f$ : all real numbers $x$
D) $\sqrt { ( x - 2 ) ^ { 2 } }$
Domain of gof: all real numbers $x$
E) $\sqrt { ( x + 2 ) ^ { 2 } }$
Domain of $g \circ f :$ all real numbers $x$

Question

Find gof and the domain of composite function. 
$f ( x ) = x ^ { 2 } + 2 , \quad g ( x ) = \sqrt { x }$
A) $( x + 2 ) ^ { 2 }$
Domain of gof: all real numbers $x$
B) $( x - 2 ) ^ { 2 }$
Domain of $g \circ f$ : all real numbers $x$
C) $\sqrt { x ^ { 2 } + 2 }$
Domain of $g \circ f$ : all real numbers $x$
D) $\sqrt { ( x - 2 ) ^ { 2 } }$
Domain of gof: all real numbers $x$
E) $\sqrt { ( x + 2 ) ^ { 2 } }$
Domain of $g \circ f :$ all real numbers $x$

Quizplus · Accepted Answer

To find $g\circ f$, we need to first plug in $f(x)$ into $g(x)$. So we have:
$$(g\circ f)(x) = g(f(x)) = g(x^2+2) = \sqrt{x^2+2}$$
The domain of $g\circ f$ is all real numbers $x$, since the square root function can take any non-negative value as input. Therefore, the correct choice is C.

Find Gof and the Domain of Composite Function f(x)=x2+2,g(x)=xf ( x ) = x ^ { 2 } + 2 , \quad g ( x ) = \sqrt { x }f(x)=x2+2,g(x)=x​

Find Gof and the Domain of Composite Function $f ( x ) = x ^ { 2 } + 2 , \quad g ( x ) = \sqrt { x }$