inverse functions (precal)?

Question

I must be going crazy because I cannot see how the book got this answer and I tried it a bunch of different ways. Essentially I just have to find the inverse of a function. (theres actually 2 I am having problems with).

The first is:

f(x)= x / (2 - x)

The book says the answer should be: 2x / (1+x)

And the similar:

f(x)= 1/(x-4)

The book says (4x+1)/x

I know there must be a step I'm missing. I've multiplied by the reciprocal, moved everything around, and ended up close, but no cigar.

Thanks in advance for the help!

Answer 1

Switch the x and y values to find the inverse.

f(x) = y = x / ( 2 - x ) so...
x = y / ( 2 - y ) and solve for y.
( 2 - y ) * x = y < distribute the x
2x - xy = y < add xy to both sides
2x = xy + y 2x = y(x + 1) < and divide by (x+1) on both sides
2x / (x+1 ) = y
which gives you the answer you found in the book.


For the second one, do the same thing.
f(x) = y = 1 / ( x - 4 )
x = 1 / ( y - 4) < multiply ( y - 4 ) on both sides
x (y - 4) = 1 y - 4 = 1 / x < add 4 on both sides
y = ( 1 / x ) + 4
This should be the answer, but whoever made the answer book make this into one big fraction.
To do this, make the 4 have a denominator of x by multiplying it by x / x. This gives you:

y = (1 / x) + (4x / x)

And now, add the two fractions

(4x + 1) / x

Hope this helped

Answer 2

f(x)= x / (2 - x)
let f(x) = y

y = x / (2 - x)

You now want to solve for x in terms of y.
y = x / (2 - x)

Take reciprocal:
1/y = (2 - x) / x

Simplify:
1/y = 2 / x - 1
1 / y + 1 = 2 / x

Convert 1 to y/y:
1 / y + y / y = 2 / x
(1 + y) / y = 2 / x

Take reciprocal:
y / (1 + y) = x / 2
2y / (1 + y) = x

Rewrite for x:
2x / (x + 1) (solution!)

Use the same methodology for the second one.