Question

True or false: If f and g are functions, then (f o g)(x) is always equal to (g o f)(x)?

Answer 1

Answer:True

Step-by-step explanation:

Answer 2

Composite functions are derived by combining multiple functions.

The statement is false

The functions are given as:

f(x) and g(x)

Where:

$\mathbf{(f\ o\ g)(x) = f(g(x))}$

and

$\mathbf{(g\ o\ f)(x) = g(f(x))}$

The true statement is that:'

$\mathbf{(f\ o\ g)(x)\ and\ (g\ o\ f)(x)}$ are not always equal

The proof is as follows:

Assume that:

$\mathbf{f(x) = 3x + 1}$

$\mathbf{g(x) = 2x - 5}$

We have:

$\mathbf{(f\ o\ g)(x) = f(g(x))}$

$\mathbf{(f\ o\ g)(x) = 3(2x - 5) + 1}$

$\mathbf{(f\ o\ g)(x) = 6x - 15 + 1}$

$\mathbf{(f\ o\ g)(x) = 6x - 14}$

Similarly:

$\mathbf{(g\ o\ f)(x) = g(f(x))}$

$\mathbf{(g\ o\ f)(x) = 2(3x + 1) - 5}$

$\mathbf{(g\ o\ f)(x) = 6x + 2 - 5}$

$\mathbf{(g\ o\ f)(x) = 6x -3}$

By comparing the results of the expressions:

$\mathbf{(f\ o\ g)(x)\ \ne \ (g\ o\ f)(x)}$

Hence, both expressions are not always equal

Read more about composite functions at:

https://brainly.com/question/10465462