Answer:
d. (g*f)(-9)=-1
Step-by-step explanation:
In this problem, we have two functions:
[tex]f(x)=\frac{3}{x}[/tex]
and
[tex]g(x)=3x[/tex]
The notation:
(f*g)(x) represents the composite function of f and g; this can be calculated by using the output of g(x) as input for f(x), in other words:
[tex](f*g)(x)=f(g(x))[/tex]
Simiarly, the notation (g*f)(x) can be calculated by using the output of f(x) as input for g(x), mathematically:
[tex](g*f)(x)=g(f(x))[/tex]
Using the definitions of f(x) and g(x), we can derive an expression for the two composite functions here:
[tex](f*g)(x)=f(g(x))=\frac{3}{g(x)}=\frac{3}{3x}=\frac{1}{x}[/tex]
And
[tex](g*f)(x)=3(f(x))=3(\frac{3}{x})=\frac{9}{x}[/tex]
Now we can analyze the given statements:
a. (f*g)(2)=2 --> FALSE, because [tex](f*g)(2)=\frac{1}{2}[/tex]
b. (g*f)(2)=0 --> FALSE, because [tex](g*f)(0)=\frac{9}{0}=\infty[/tex]
c.(f*g)(9)=1 --> FALSE, because [tex](f*g)(9)=\frac{1}{9}[/tex]
d. (g*f)(-9)=-1 --> TRUE, because [tex](g*f)(-9)=\frac{9}{-9}=-1[/tex]
