Inverse Functions

**Tom G** · September 12th 2007, 04:44 AM

I've got two functions [f(x) and g(x)] which I need to find the inverse of [f-(x) and g-(x)], can someone please explain the steps taken when finding the inverse of funtions.

1. f(x)= the square root of 'x' (i don't have a square root button on my keyboard)

2. g(x)= 2x+1

I would appreciate if someone could show each step they have taken to get the answer.

**topsquark** · September 12th 2007, 05:07 AM

Originally Posted by Tom G

I've got two functions [f(x) and g(x)] which I need to find the inverse of [f-(x) and g-(x)], can someone please explain the steps taken when finding the inverse of funtions.

1. f(x)= the square root of 'x' (i don't have a square root button on my keyboard)

2. g(x)= 2x+1

I would appreciate if someone could show each step they have taken to get the answer.

The general method is to take a function $y = f(x)$ , reverse the roles of x and y ( $x = f(y)$ , then solve this equation for y ( $y = g(x) = f^{-1}(x)$ .)

When doing this you need to be careful about domains and ranges. (The range of the function is the domain of the inverse function and visa versa.)

For example:
1) $y = f(x) = \sqrt{x}$
Switch the roles of x and y: $x = \sqrt{y}$
Solve for y: $y = x^2$

Thus $f^{-1}(x) = x^2$ .

Note, though, that since f(x) is only defined on $[0, \infty )$ that the inverse function is only defined on $[0, \infty )$ rather than $(-\infty, \infty )$ .

2) $g(x) = 2x + 1$ .
You do this one. I get $g^{-1}(x) = \frac{x - 1}{2}$ .

-Dan

**red_dog** · September 12th 2007, 05:14 AM

If $f:A\to B$ and $f(x)=y$ then $f^{-1}:B\to A$ and $f^{-1}(y)=x$ .
So, to find $f^{-1}$ you have to find $y$ as a function of $x$ .

1) $f:[0,\infty)\to[0,\infty),f(x)=\sqrt{x}$ .
$\sqrt{x}=y$
Square both sides:
$x=y^2$ So, $f^{-1}(y)=y^2$
Now, change the letter for the variable:
$f^{-1}:[0,\infty)\to[0,\infty), \ f^{-1}(x)=x^2$

Now, can you solve 2)?

**Tom G** · September 12th 2007, 05:22 AM

Thanks, I've now managed to do g(x) and got the same answer as topsquark.

I've got:

g(x)=2x+1

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

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

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