+0

# Let Find the function k(x) such that f is its own inverse.

-1
247
5
+979

Let$$f(x) = \begin{cases} k(x) &\text{if }x>2, \\ 2+(x-2)^2&\text{if }x\leq2. \end{cases}$$Find the function k(x) such that f is its own inverse.

Apr 16, 2020

#1
+111983
0

I showed you how to do these yesterday.

Did you attempt to learn?

Apr 16, 2020
#2
+979
0

I did, in fact, attempt to learn.  But I had no idea how you did it so I didn't learn.  Thank you for you positive comment that made me feel sooo good.

qwertyzz  Apr 16, 2020
#3
+111983
+1

It really showed ME how much you wanted to understand!

For the benefit of other readers: quertyzz gave me NO feedback.

Melody  Apr 16, 2020
edited by Melody  Apr 16, 2020
#4
+111983
+1

When x=2

f(x)= 2 +0 = 2

(2,2) is on the line y=x.

The inverse of a function is its reflection over the line y=x

you can find it by swapping the x with the y  (although occasionally there can be a problem.)

so

if a function is

y=2+(x-2)^2

then the inverse will be

x=2+(y-2)^2

which simplifies as follows

$$x=2+(y-2)^2\\ x=2+(y^2-4y+4)\\ x=y^2-4y+6\\ x-6=y^2-4y\\ x-6+4=y^2-4y+4\\ x-2=(y-2)^2\\ \pm\sqrt{x-2}=y-2\\ y=2\pm\sqrt{x-2}\\$$

Ok, I have hit a slight problem.  Which one will I choose?  It is not going to be both.

x>2 and x-2 must be bigger than 0 so that is always true, so that didn't help.

Lets consider the original function some more

$$f(x) =(x-2)^2 +2 \qquad where \qquad x\leq2$$

This is the left side of a concave up parabola with the vertex at (2,2)

So it is always going to be above the line y=x.

So the inverse will always be below the line y=x

This will only be true of the negative case.

This can also be seen from the graph x=(y-2)^2+2    (part of the working above)

which is a sideways parabola and you want the bottom half so you want the neg square root.

$$f(x) = \begin{cases} k(x) &\text{if }x>2, \\ 2+(x-2)^2&\text{if }x\leq2. \end{cases}\\ \bf{k(x)=2-\sqrt{x-2}}$$

check:

Here is the graph:  https://www.desmos.com/calculator/9ntcun0vs6

Given that there were complications, someone may be able to show you a more straight forward method. Not sure about that.

LaTex:

x=2+(y-2)^2\\
x=2+(y^2-4y+4)\\
x=y^2-4y+6\\
x-6=y^2-4y\\

x-6+4=y^2-4y+4\\

x-2=(y-2)^2\\

\pm\sqrt{x-2}=y-2\\

y=2\pm\sqrt{x-2}\\

Apr 16, 2020
#5
+111983
0

Did you learn anything that time qwerty?

I see you are so grateful when people try to help you that much of the time you cannot even be bothered responding.

Apr 17, 2020