## anonymous one year ago 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.

1. mathstudent55

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

2. UsukiDoll

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

3. mathstudent55

Is this f(x)? $$f(x) = \dfrac{x - 8}{x + 7}$$ Is g(x) $$g(x) = \dfrac{-7x - 8}{x - 1}$$ ?

4. anonymous

Yesss @mathstudent55

5. mathstudent55

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.

6. mathstudent55

Ok. Let's do it. Functions f(x) and g(x) are inverses if f(g(x)) = g(f(x)) = x

7. mathstudent55

$$f(x) = \dfrac{x - 8}{x + 7}$$ 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 $$\dfrac{-7x - 8}{x - 1}$$

8. mathstudent55

$$\large f(x) = \dfrac{x - 8}{x + 7}$$ $$\large f(g(x)) = f\left(\dfrac{-7x - 8}{x - 1}\right)$$ $$\large f(g(x))\Large = \dfrac{\frac{-7x - 8}{x - 1} - 8}{\frac{-7x - 8}{x - 1} + 7}$$ You see, where there was an x in the f(x) function, we now have what g(x) is equal to.

9. mathstudent55

Now we need to simplify that fraction.

10. mathstudent55

$$\large f(g(x))\Large = \dfrac{\left( \frac{-7x - 8}{x - 1} - 8 \right)(x - 1)}{\left( \frac{-7x - 8}{x - 1} + 7 \right) (x - 1)}$$ $$\large f(g(x))\Large = \dfrac{-7x - 8 - 8(x - 1)}{-7x - 8 + 7 (x - 1)}$$ $$\large f(g(x))\Large = \dfrac{-7x - 8 - 8x + 8}{-7x - 8 + 7x - 7}$$ $$\large f(g(x))\Large = \dfrac{-15x}{-15}$$ $$\large f(g(x))\Large = x$$

11. mathstudent55

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

12. mathstudent55

$$\large g(x) = \dfrac{-7x - 8}{x - 1}$$ $$\large f(x) = \dfrac{x - 8}{x + 7}$$ Now we replace the x on the right side of the g(x) function with what f(x) is equal to, $$\dfrac{x - 8}{x + 7}$$ $$\large g(f(x)) \Large = \dfrac{-7 \left( \frac{x - 8}{x + 7} \right) -8}{ \frac{x - 8}{x + 7} - 1}$$ $$\large g(f(x))\Large = \dfrac{\left[ -7 \left( \frac{x - 8}{x + 7} \right) -8 \right](x + 7)}{\left[ \frac{x - 8}{x + 7} - 1 \right](x + 7)}$$ $$\large g(f(x))\Large = \dfrac{-7 (x - 8) -8 (x + 7)}{x - 8 - (x + 7)}$$ $$\large g(f(x))\Large = \dfrac{-7x + 56 -8x - 56)}{x - 8 - x - 7}$$ $$\large g(f(x))\Large = \dfrac{-15x}{- 15 }$$ $$\large g(f(x))\Large = x$$

13. mathstudent55

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.

14. anonymous

thank you sooooooo much @mathstudent55

15. mathstudent55

You are very welcome.