If f(x)=3/x and g(x)=3x, which of the following statements is true?
a. (f*g)(2)=2
b. (g*f)(2)=0
c.(f*g)(9)=1
d. (g*f)(-9)=-1

d. (g*f)(-9)=-1

Step-by-step explanation:

In this problem, we have two functions:

and

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:

Simiarly, the notation (g*f)(x) can be calculated by using the output of f(x) as input for g(x), mathematically:

Using the definitions of f(x) and g(x), we can derive an expression for the two composite functions here:

And

Now we can analyze the given statements:

a. (f*g)(2)=2  --> FALSE, because

b. (g*f)(2)=0  --> FALSE, because

c.(f*g)(9)=1  --> FALSE, because

d. (g*f)(-9)=-1 --> TRUE, because

a negative exponent is translated as follows:

so, we have

answer:

c. the third graph