Question

If f(x)=3/x and g(x)=3x, which of the following statements is true?

Answer 1

Answer:

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

Step-by-step explanation:

In this problem, we have two functions:

$f(x)=\frac{3}{x}$

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:

$(f*g)(x)=f(g(x))=\frac{3}{g(x)}=\frac{3}{3x}=\frac{1}{x}$

And

$(g*f)(x)=3(f(x))=3(\frac{3}{x})=\frac{9}{x}$

Now we can analyze the given statements:

a. (f*g)(2)=2  --> FALSE, because $(f*g)(2)=\frac{1}{2}$

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

c.(f*g)(9)=1  --> FALSE, because $(f*g)(9)=\frac{1}{9}$

d. (g*f)(-9)=-1 --> TRUE, because $(g*f)(-9)=\frac{9}{-9}=-1$