anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
mathstudent55
  • mathstudent55
I don't understand what the functions are. Can you use parentheses to show the numerator and denominator?
UsukiDoll
  • UsukiDoll
agree I can't figure it out either. Can you draw the functions?
mathstudent55
  • mathstudent55
Is this f(x)? \(f(x) = \dfrac{x - 8}{x + 7} \) Is g(x) \(g(x) = \dfrac{-7x - 8}{x - 1} \) ?

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anonymous
  • anonymous
Yesss @mathstudent55
mathstudent55
  • 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.
mathstudent55
  • mathstudent55
Ok. Let's do it. Functions f(x) and g(x) are inverses if f(g(x)) = g(f(x)) = x
mathstudent55
  • 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 \(\dfrac{-7x - 8}{x - 1}\)
mathstudent55
  • 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.
mathstudent55
  • mathstudent55
Now we need to simplify that fraction.
mathstudent55
  • mathstudent55
\(\large f(g(x))\Large = \dfrac{\left( \frac{-7x - 8}{x - 1} - 8 \right)(x - 1)}{\left( \frac{-7x - 8}{x - 1} + 7 \right) (x - 1)}\) \(\large f(g(x))\Large = \dfrac{-7x - 8 - 8(x - 1)}{-7x - 8 + 7 (x - 1)}\) \(\large f(g(x))\Large = \dfrac{-7x - 8 - 8x + 8}{-7x - 8 + 7x - 7}\) \(\large f(g(x))\Large = \dfrac{-15x}{-15}\) \(\large f(g(x))\Large = x\)
mathstudent55
  • mathstudent55
That shows that f(g(x)) = x. Now we need to show that g(f(x)) = x
mathstudent55
  • 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, \(\dfrac{x - 8}{x + 7} \) \(\large g(f(x)) \Large = \dfrac{-7 \left( \frac{x - 8}{x + 7} \right) -8}{ \frac{x - 8}{x + 7} - 1}\) \(\large g(f(x))\Large = \dfrac{\left[ -7 \left( \frac{x - 8}{x + 7} \right) -8 \right](x + 7)}{\left[ \frac{x - 8}{x + 7} - 1 \right](x + 7)} \) \(\large g(f(x))\Large = \dfrac{-7 (x - 8) -8 (x + 7)}{x - 8 - (x + 7)} \) \(\large g(f(x))\Large = \dfrac{-7x + 56 -8x - 56)}{x - 8 - x - 7} \) \(\large g(f(x))\Large = \dfrac{-15x}{- 15 } \) \(\large g(f(x))\Large = x\)
mathstudent55
  • 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.
anonymous
  • anonymous
thank you sooooooo much @mathstudent55
mathstudent55
  • mathstudent55
You are very welcome.

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