What are the inverses of the following equations?

Question

1) F(x)=12-4x
2) F(x)=1/x
3) F(x)=(x-1)/(2-x)

Answer 1

To find the inverse of a function, just swap the places of "x" and "F(x)" and then solve for F(x).

1) x=12-4F(x)
x-12=-4F(x)
F(x)=(12-x)/4

2) x=1/F(x)
F(x)=1/x --it's its own inverse.

3) x=(F(x)-1)/(2-F(x))
x(2-F(x))=(F(x)-1)
2x-xF(x)=F(x)-1
2x+1=F(x)+xF(x)
2x+1=F(x)(1+x)
F(x)=(2x+1)/(1+x)

Answer 2

General method of solving inverses.
(a) Let y = F(x),
(b) Switch the x and y variables.
(c) Solve for y.

1) F(x) = 12 - 4x

First, we assign y = F(x).

y = 12 - 4x

Now, we switch the x and y variables...

x = 12 - 4y

And now we solve for y.

4y = - x + 12
y = (-1/4)x + 3

We can't stop here. What we would do now is say, as our concluding statement,

Therefore, F^(-1)(x) = (-1/4)x + 3.
{You would write "f inverse" as putting the -1 symbol as a power between the F and the x.}

2. F(x) = 1/x.

Again, let y = F(x). Then
y = 1/x, Switch the variables,
x = 1/y, and solve for y

y = 1/x.
Therefore, F^(-1)(x) = 1/x
{This implies that the inverse is itself. All functions symmetric across the origin are inverses of themselves}

3) F(x) = (x - 1)/(2 - x)

Let y = F(x). Then

y = (x - 1)/(2 - x)

Switch the variables.

x = (y - 1)/(2 - y)

And then solve for y. First, we have to multiply both sides by
(2 - y).

x(2 - y) = y - 1. Expand the left hand side, to get
2x - xy = y - 1. Now, move all y terms to the left hand side, and everything else to the right hand side.

-xy - y = -2x - 1. Factor a y out of the left hand side, to get
y(-x - 1) = -2x - 1. Now, divide both sides to isolate y.

y = (-2x - 1) / (-x - 1)

We can rewrite this slightly, by factoring (-1) out of the top and bottom,

y = [(-1)(2x + 1)]/[(-1)(x+1)]

Which means we can effectively cancel out the -1, to obtain

y = (2x + 1)/(x + 1)

And don't forget our concluding statement:

Therefore,
F^(-1)(x) = (2x + 1)/(x + 1)

Answer 3

I'm going to let y be the inverse:

1) x = 12 - 4y then solve for y (easy)

2) x = 1/y then solve for y (easy also)

3) x = (y-1) / (2-y) slightly challenging

Multiply both sides by 2-y so you get

x(2-y) = (y-1)

2x-xy=y-1

Move all terms with y to one side and all others to the other side:

2x +1 = y + xy

2x +1 = y (1 + x)

Divide by (1+x)

So y = (2x+1)/(1+x)

Answer 4

Change f(x) to x and x to f(x) and solve for f(x).

1)

x = 12 - 4f(x)
x - 12 = -4f(x)
f(x) = -1/4 x + 3

2)

x = 1/f(x)
f(x) = 1/x

3)

x = (f(x) - 1)/(2 - f(x))
x(2 - f(x)) = f(x) - 1
2x - xf(x) = f(x) - 1
2x + 1 = f(x) + xf(x) = f(x)(1 + x)
f(x) = (2x + 1)/(1 + x)

Answer 5

Leave x on one side.

1) y=12-4x
4x=12-y
x=(12-y)/4

F^-1(x)= (12-x)/4=3-x/4

2) xy=1
x=1/y
F^-1(x)=1/x

3) cross multiplication
2y-xy=x-1
2y+1=xy+x

x(y+1)= 2y+1

x=(2y+1)/(y+1)

F^-1(x)=(2x+1)/(x+1)