Answer:
[tex]g(x)=x+6[/tex] is the answer
given
[tex]h(x)=4\sqrt{x+7}[/tex] and [tex]f(x)=4\sqrt{x+1}[/tex].
Step-by-step explanation:
[tex]h(x)=(f \circ g)(x)[/tex]
[tex]h(x)=f(g(x))[/tex]
Inputting the given function for h(x) into the above:
[tex]4\sqrt{x+7}=f(g(x))[/tex]
Now we are plugging in g(x) for x in the expression for f which is [tex]4\sqrt{x+1}[/tex] which gives us [tex]4\sqrt{g(x)+1}[/tex]:
[tex]4\sqrt{x+7}=4\sqrt{g(x)+1}[/tex]
We want to solve this for g(x).
If you don't like the looks of g(x) (if you think it is too daunting to look at), replace it with u and solve for u.
[tex]4\sqrt{x+7}=4\sqrt{u+1}[/tex]
Divide both sides by 4:
[tex]\sqrt{x+7}=\sqrt{u+1}[/tex]
Square both sides:
[tex]x+7=u+1[/tex]
Subtract 1 on both sides:
[tex]x+7-1=u[/tex]
Simplify left hand side:
[tex]x+6=u[/tex]
[tex]u=x+6[/tex]
Remember u was g(x) so you just found g(x) so congratulations.
[tex]g(x)=x+6[/tex].
Let's check it:
[tex](f \circ g)(x)[/tex]
[tex]f(g(x))[/tex]
[tex]f(x+6)[/tex] I replace g(x) with x+6 since g(x)=x+6.
[tex]4\sqrt{(x+6)+1}[/tex] I replace x in f with (x+6).
[tex]4\sqrt{x+6+1}[/tex]
[tex]4\sqrt{x+7}[/tex]
[tex]h(x)[/tex]
The check is done. We have that [tex](f \circ g)(x)=h(x)[/tex].
