Solving (a)
The two functions we have are:
[tex]\begin{gathered} f(x)=3x+3 \\ g(x)=x^2 \end{gathered}[/tex]
We are asked to find the composite function:
[tex](f\circ g)(x)[/tex]
Step 1. The definition of a composite function is:
[tex](h\circ k)(x)=h(k(x))[/tex]
In this case:
[tex](f\circ g)(x)=f(g(x))[/tex]
This means to plug the g(x) expression into the value of x of the f(x) function.
Step 2. Substituitng g(x) as the value for x in f(x):
[tex](f\circ g)(x)=f(g(x))=4(x^2)+3[/tex]
Simplifying:
[tex](f\circ g)(x)=\boxed{4x^2+3}[/tex]
Step 3. We also need to find the domain of this composite function.
The domain of a function is the possible values that the x-variable can take. In this case, there would be no issues with any x value that we plug as the x-value. Therefore, the domain is all real numbers.
The domain of fog is all real numbers.
Answer:
[tex](f\circ g)(x)=\boxed{4x^2+3}[/tex]
The domain of fog is all real numbers.
