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

Question

asked Dec 24, 2021 34.2k views

1 Answer

Redskull · Answer 1 · 2021-12-30T02:33:16+0000

Answer:

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

Explanation:

In this problem, we have two functions:

f(x)=(3)/(x)

and

g(x)=3x

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:

(f*g)(x)=f(g(x))

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

(g*f)(x)=g(f(x))

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

(f*g)(x)=f(g(x))=(3)/(g(x))=(3)/(3x)=(1)/(x)

And

(g*f)(x)=3(f(x))=3((3)/(x))=(9)/(x)

Now we can analyze the given statements:

a. (f*g)(2)=2 --> FALSE, because
(f*g)(2)=(1)/(2)

b. (g*f)(2)=0 --> FALSE, because
$(g*f)(0)=(9)/(0)=\infty$

c.(f*g)(9)=1 --> FALSE, because
(f*g)(9)=(1)/(9)

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