# Find the inverse function, if exists..

• Apr 8th 2009, 06:02 AM
TWiX
Find the inverse function, if exists..
hello
am stopped at this problem -_-

let $\displaystyle f(x)=2x^2-8x$ where x>2
Find a formula for the inverse function of it, if exists.

My solution:
f(x)=4x-..... so f is increas..so its 1-1funct..so inverse is exist.

its easy to prove that the inverse function is exist..=)

But the problem is finding the inverse function
ok let $\displaystyle y=2x^2-8x$ ..
how to find the x in terms of y !!
i tried to get the natural logarthmic of both sides but it failed :S

any help?!
• Apr 8th 2009, 06:10 AM
Jester
Quote:

Originally Posted by TWiX
hello
am stopped at this problem -_-

let $\displaystyle f(x)=2x^2-8x$ where x>2
Find a formula for the inverse function of it, if exists.

My solution:
f(x)=4x-..... so f is increas..so its 1-1funct..so inverse is exist.

its easy to prove that the inverse function is exist..=)

But the problem is finding the inverse function
ok let $\displaystyle y=2x^2-8x$ ..
how to find the x in terms of y !!
i tried to get the natural logarthmic of both sides but it failed :S

any help?!

Switch $\displaystyle x\; \text{and}\; y$, complete the square in y and then solve for y.
• Apr 8th 2009, 06:11 AM
thelostchild
complete the square on the right hand side to give
$\displaystyle y=2x^2-8x$

$\displaystyle \frac{y}{2}=x^2-4x$

$\displaystyle \frac{y}{2}=(x-2)^2-4$

I assume you can take it from there!
• Apr 8th 2009, 06:17 AM
Prove It
Quote:

Originally Posted by TWiX
hello
am stopped at this problem -_-

let $\displaystyle f(x)=2x^2-8x$ where x>2
Find a formula for the inverse function of it, if exists.

My solution:
f`(x)=4x-..... so f is increas..so its 1-1funct..so inverse is exist.

its easy to prove that the inverse function is exist..=)

But the problem is finding the inverse function
ok let $\displaystyle y=2x^2-8x$ ..
how to find the x in terms of y !!
i tried to get the natural logarthmic of both sides but it failed :S

any help?!

$\displaystyle f(x)$ IS one to one, but only because the domain has been restricted to all x-values greater than the turning point, not for the reason you stated.

The domain of f is $\displaystyle x > 2$, so the range is $\displaystyle y > 0$, since the function is increasing when $\displaystyle x>2$...

Note that for an inverse function, the domain and range swap, which implies that the x and y values swap.

So $\displaystyle f^{-1}(x): x = 2y^2 - 8y$

$\displaystyle \frac{1}{2}x = y^2 - 4y$

$\displaystyle \frac{1}{2}x + \left(-\frac{4}{2}\right)^2 = y^2 - 4y + \left(-\frac{4}{2}\right)^2$

$\displaystyle \frac{1}{2}x + 4 = (y - 2)^2$

Can you solve for y now? Remember that the domain of $\displaystyle f^{-1}$ is $\displaystyle x>0$ since $\displaystyle \textrm{dom}f^{-1} = \textrm{ran}f$.
• Apr 8th 2009, 06:52 AM
TWiX
Get it now ..
Need to solve the equation for x without using the complete of the square ??!
• Apr 8th 2009, 07:19 AM
stapel
Quote:

Originally Posted by TWiX
let $\displaystyle f(x)=2x^2-8x$ where x>2

Find a formula for the inverse function of it, if exists.

Use the standard process:

Rename "f" as "y": y = 2x^2 - 8x

Solve this for "x=". In this case, use the Quadratic Formula:

. . . . .$\displaystyle 2x^2\, -\, 8x\, -\, y\, =\, 0$

. . . . .$\displaystyle x\, =\, \frac{-(-8)\, \pm\, \sqrt{(-8)^2\, -\, 4(2)(-y)}}{2(2)}\, =\, \frac{8\, \pm\, \,\sqrt{64\, +\, 8y}}{4}$

...and so forth. In this case, you were given the restriction of x > 2, so you know you need only the positive square root:

. . . . .$\displaystyle x\, =\, 2\, +\, \frac{\sqrt{16\, +\, 2y}}{2}$

Then continue with the usual steps: Swap "x" and "y", and rename the new "y" as f-inverse.

(Wink)
• Apr 8th 2009, 08:43 AM
TWiX
Thank you!
imo i think that solution is more easy than the first one -_-

Thanks.