Question

Consider the two functions shown below.

Answer 1

Answer: Option B

Then The functions f(x) and g(x) are inverses because $f(g(x))=g(f(x))=x$

Step-by-step explanation:

To answer this question we must make the composition of f (x) and g (x)

We found

$f (g(x))$

We know that

$f(x) = 2x-2\\\\g(x) = \frac{1}{2}x + 1$

By definition if two functions f and g are inverses then it follows that:

$f(g(x)) = g(f(x)) = x$

So if $f(g(x)) = x$ f and g are inverse

To find  $f(g(x))$ enter the function g(x) within  the function f(x) as shown below

$f(g(x))= 2( \frac{1}{2}x + 1)-2\\\\f(g(x))= x + 2 -2\\\\f(g(x))= x$

Now

$g(f(x))= \frac{1}{2}(2x-2) + 1\\\\g(f(x))= x-1 + 1\\\\g(f(x))= x$

Observe that

$f(g(x))=g(f(x))=x$

Then The functions f(x) and g(x) are inverses because $f(g(x))=g(f(x))=x$

Answer 2

ANSWER

B. The functions f(x) and g(x) because f(g(x))=g(f(x))=x

EXPLANATION

The given functions are:

$f(x) = 2x - 2$

and

$g(x) = \frac{1}{2}x + 1$

If f(x) and g(x) are inverses, then

f(g(x))=x

$f(g(x))=f( \frac{1}{2} x + 1)$

$f(g(x))=2( \frac{1}{2}x + 1) - 2$

Expand the parenthesis to obtain,

$f(g(x))=x + 2- 2$

$f(g(x))=x$

Also,

$g(f(x)) = g(2x - 2)$

$g(f(x)) = \frac{1}{2} (2x - 2) + 1$

$g(f(x)) = x - 1+ 1$

$g(f(x)) = x$

Hence f(x) and g(x) are inverses