pre calc question

Question

f(x)-= x+5/x+8

f^-1(-8)=

Answer 1

You could take the inverse of f(x) and then plug in x = -8 to find the solution. To do this:

1) replace f(x) with "y": y = (x + 5)/(x + 8)

2) swap the "x's" and "y's": x = (y + 5)/(y + 8)

3) solve for "y":

x(y + 8) = y + 5

xy + 8x = y + 5

xy - y = 5 - 8x

y(x - 1) = (5 - 8x)

y = (5 - 8x)/(x - 1)

4) rewrite as f^-1(x):

f^-1(x):= (5 - 8x)/(x - 1)

5) evaluate at x = -8:

f^-1(-8):= (5 - 8(-8))/((-8) - 1) = 69/-9 = -23/3

But, probably easier, is to realize that if (a, b) is a point on f(x) then (b, a) will be a point on f^-1(x). So, we are given "b" because the question wants to know what f^-1(-8) is. We are just trying to find "a". In other words, we want to know what the "x" value is when f(x) = -8 because that will be the same as the "y" value of

f^-1(-8).

So make f(x) = -8 and solve for "x":

-8 = (x + 5)/(x + 8)

-8(x + 8) = x + 5

-8x - 64 = x + 5

-64 - 5 = 8x + x

-69 = 9x

x = -69/9 = -23/3

So either way, f^-1(-8) = -23/3

Answer 2

If f(a) = b, then f^-1(b) = a

So, first find x so that f(x) = -8:

(x+5) / (x+8) = -8

x+5 = -8x - 64

9x = -69

x = -69/9

So, since f(-69//9) = -8, f^-1(-8 ) = -69/9