Expert: Paul Klarreich - 3/14/2007

Question
How do i find the inverse of F(x)=2x-7?
f(x)=3-2x/5x+1?
f(x)=3/2x^3+1? and
f(x)=3x+4?

Answer
Questioner:   Brandy
Category:  Calculus
 
Subject:  Calculus
Question:  How do i find the inverse of F(x)=2x-7?
f(x)=3-2x/5x+1?
f(x)=3/2x^3+1? and
f(x)=3x+4?

....................................
Hi, Brandy,

Please, when you send questions involving fractions, PARENTHESIZE.

3-2x/5x+1  could mean:
   2x
3 - -- + 1
   5x

I don't think you meant that; I think you meant to write:

(3-2x)/(5x+1)

which is the way to write it.

.........................
These look like four separate questions, but the routine is the same in each case, although details might vary:

1. Write the function as  y = f(x).
2. Swap  the letters  x and y.
3. Solve for  y in terms of x.  That's your inverse function, IF IT EXISTS.

You have to make sure it does, and that might involve some tricky stuff about the domain and range, but maybe we won't have to.


1.  F(x)=2x-7
Write  y = 2x - 7
Swap:  x = 2y - 7
Solve: x + 7 = 2y
    x + 7
y = -------
      2
That's the inverse.
........................
2. f(x)= (3-2x)/(5x+1)

Write:  y = (3-2x)/(5x+1)
Swap:   x = (3-2y)/(5y+1)
Solve:

5xy + x = 3 - 2y
5xy + 2y = 3 - x
y(5x + 2) = 3 - x

y = (3 - x)/(5x + 2)
.....................

3. f(x) = 3/2x^3+1?

I am going to assume you mean  f(x) = (3/2)x^3 + 1

Write:  y = (3/2)x^3 + 1
Swap:   x = (3/2)y^3 + 1
Solve:  x - 1 = (3/2)y^3

    y^3 = 2(x - 1)/3
    y = [2(x - 1)/3]^1/3 or
    y = cuberootof(2(x - 1)/3)


4. f(x)=3x+4

I'll leave this one to you -- it is the same as example 1 with different numbers.