Specify the domain of $( g \circ f ) ( x )$ for the pair of functions $f ( x ) = 3 x$ and $g ( x ) = 8 x - 7$ .
A) $( - \infty , 0 )$
B) $( - \infty , 0 ) \cup ( 0 , \infty )$
C) $( 0 , \infty )$
D) $( - \infty , \infty )$
E) $( - \infty , 0 ]$

Question

Quizplus · Accepted Answer

The domain of $f(x)$ is $(-\infty,\infty)$ because any real number can be inputted into the function.
The range of $f(x)$ is also $(-\infty,\infty)$ because any real number can be outputted by the function.
The domain of $g(x)$ is $(-\infty,\infty)$ because any real number can be inputted into the function.
The range of $g(x)$ is $(-\infty,\infty)$ because any real number can be outputted by the function.
The domain of $(g\circ f)(x) = g(f(x))$ is the set of all real numbers that can be inputted into $f$, such that the output of $f$ can be inputted into $g$. Since the range of $f$ is $(-\infty,\infty)$ and the domain of $g$ is also $(-\infty,\infty)$, we can input any real number into $f$ and obtain any real output from $f$, which means any output of $f$ can be inputted into $g$. Therefore, the domain of $(g\circ f)(x)$ is $(-\infty,\infty)$.

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