Question

PLEASE HELP!!! I HAVE NO IDEA WHAT IM DOING

Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. f(x) = quantity x-8/ x+7. and gx) = -7x - 8/x-1.

Answer 1

I don't understand what the functions are.
Can you use parentheses to show the numerator and denominator?

Answer 2

agree I can't figure it out either. Can you draw the functions?

Answer 3

Is this f(x)?

\(f(x) = \dfrac{x - 8}{x + 7} \)

Is g(x)

\(g(x) = \dfrac{-7x - 8}{x - 1} \) ?

Answer 4

Yesss @mathstudent55

Answer 5

Ok. Now we can start.
Before we start, next time do it this way. Then there won't be any confusion:

f(x) = (x - 8)/(x+7)
g(x) = (-7x - 8)/(x - 1)

Also, you can draw the functions or use the Latex editor.

Answer 6

Ok. Let's do it.

Functions f(x) and g(x) are inverses if f(g(x)) = g(f(x)) = x

Answer 7

\(f(x) = \dfrac{x - 8}{x + 7}\)

f(g(x)) means to evaluate f(x) using g(x)

\(g(x) = \dfrac{-7x - 8}{x - 1} \), so we replace x of f(x) with \(\dfrac{-7x - 8}{x - 1}\)

Answer 8

\(\large f(x) = \dfrac{x - 8}{x + 7} \)

\(\large f(g(x)) = f\left(\dfrac{-7x - 8}{x - 1}\right)\)

\(\large f(g(x))\Large = \dfrac{\frac{-7x - 8}{x - 1} - 8}{\frac{-7x - 8}{x - 1} + 7}\)

You see, where there was an x in the f(x) function, we now have what g(x) is equal to.

Answer 9

Now we need to simplify that fraction.

Answer 10

\(\large f(g(x))\Large = \dfrac{\left( \frac{-7x - 8}{x - 1} - 8 \right)(x - 1)}{\left( \frac{-7x - 8}{x - 1} + 7 \right) (x - 1)}\)

\(\large f(g(x))\Large = \dfrac{-7x - 8 - 8(x - 1)}{-7x - 8 + 7 (x - 1)}\)

\(\large f(g(x))\Large = \dfrac{-7x - 8 - 8x + 8}{-7x - 8 + 7x - 7}\)

\(\large f(g(x))\Large = \dfrac{-15x}{-15}\)

\(\large f(g(x))\Large = x\)

Answer 11

That shows that f(g(x)) = x.

Now we need to show that g(f(x)) = x

Answer 12

\(\large g(x) = \dfrac{-7x - 8}{x - 1} \)

\(\large f(x) = \dfrac{x - 8}{x + 7} \)

Now we replace the x on the right side of the g(x) function with what f(x) is equal to, \(\dfrac{x - 8}{x + 7} \)

\(\large g(f(x)) \Large = \dfrac{-7 \left( \frac{x - 8}{x + 7} \right) -8}{ \frac{x - 8}{x + 7} - 1}\)

\(\large g(f(x))\Large = \dfrac{\left[ -7 \left( \frac{x - 8}{x + 7} \right) -8 \right](x + 7)}{\left[ \frac{x - 8}{x + 7} - 1 \right](x + 7)} \)

\(\large g(f(x))\Large = \dfrac{-7 (x - 8) -8 (x + 7)}{x - 8 - (x + 7)} \)

\(\large g(f(x))\Large = \dfrac{-7x + 56 -8x - 56)}{x - 8 - x - 7} \)

\(\large g(f(x))\Large = \dfrac{-15x}{- 15 } \)

\(\large g(f(x))\Large = x\)

Answer 13

Now we have also shown that g(f(x)) = x.

Since we now know that f(g(x)) = g(f(x)) = x, we have confirmed that the functions f(x) and g(x) are inverses of each other.

Answer 14

thank you sooooooo much @mathstudent55

Answer 15

You are very welcome.