Determine whether the statements are True or False. Justify your answer with an explanation. If f(x) = x + 1 and g(x) = 6x, then (f ○ g)(x) = (g ○ f)(x).

If you are given two functions f(x) and g(x), you can calculate (f ○ g)(x) if and only if the range of g is a subset of the domain of f.

Problem 1

Explanation:

The notation (f o g)(x) means f( g(x) ). Here g(x) is the inner function.

So,

f(x) = x+1

f( g(x) ) = g(x) + 1 replace every x with g(x)

f( g(x) ) = 6x+1 ... plug in g(x) = 6x

(f o g)(x) = 6x+1

Now let's flip things around

g(x) = 6x

g( f(x) ) = 6*( f(x) ) replace every x with f(x)

g( f(x) ) = 6(x+1) plug in f(x) = x+1

g( f(x) ) = 6x+6

(g o f)(x) = 6x+6

This shows that (f o g)(x) = (g o f)(x)  is a false equation for the given f(x) and g(x) functions.

Problem 2

Explanation:

Let's say that g(x) produced a number that wasn't in the domain of f(x). This would mean that f( g(x) ) would be undefined.

For example, let

f(x) = 1/(x+2)

g(x) = -2

The g(x) function will always produce the output -2 regardless of what the input x is. Feeding that -2 output into f(x) leads to 1/(x+2) = 1/(-2+2) = 1/0 which is undefined.

So it's important that the outputs of g(x) line up with the domain of f(x). Outputs of g(x) must be valid inputs of f(x).

d is the correct answer

step-by-step explanation:

x= 8 because 3x = 3 * 8, and 3*8 = 24