Question

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = the quantity x minus seven divided by the quantity x plus three. and g(x) = quantity negative three x minus seven divided by quantity x minus one.

Answer 1

Answer with Step-by-step explanation:

We are given that

`f(x)=(x-7)/(x+3)`

`g(x)=(-3x-7)/(x-1)`

We have to confirm that f and g are inverses by showing that f(g(x))=x=g(f(x))

`f(g(x))=f((-3x-7)/(x-1))`

`f(g(x))=((-3x-7)/(x-1)-7)/((-3x-7)/(x-1)+3)`

`f(g(x))=((-3x-7-7x+7)/(x-1))/((-3x-7+3x-3)/(x-1))`

`f(g(x))=(-10x)/(x-1)* (x-1)/(-10)`

`f(g(x))=x`

`g(f(x))=g((x-7)/(x+3))`

`g(f(x))=(-3((x-7)/(x+3))-7)/((x-7)/(x+3)-1)`

`g(f(x))=((-3x+21-7x-21)/(x+3))/((x-7-x-3)/(x+3))`

`g(f(x))=((-10x)/(x+3))/((-10)/(x+3))`

`g(f(x))=(-10x)/(x+3)* (x+3)/(-10)=x`

Hence, f(g(x))=g(f(x))=x

Therefore, f and g are inverses.

Answer 2

Answer: The prove is mentioned below.

Explanation:

Here the given functions are,

`f(x)=(x-7)/(x+3)`

`g(x)=(-3x-7)/(x-1)`

We have to prove that: f(g(x)) = x and g(f(x)) = x.

Since
`f(g(x))= ((-3x-7)/(x-1) -1)/((-3x-7)/(x-1) +3)`

⇒
`f(g(x))= (-3x-7-7x+7)/(-3x-7+3x-3)`

⇒
`f(g(x))= (-4x)/(-4)`

⇒
`f(g(x))= x`

Now,
`g(f(x))= (-3(x-7)/(x+3) -7)/((x-7)/(x+3) -1)`

⇒
`g(f(x))= (-3x+21-7x-21)/(x-7-x-3)`

⇒
`g(f(x))= (-10x)/(-10)`

⇒
`g(f(x))= x`