Question

I do not understand inverse functions exactly so if you could explain this problem that would be great

If f(x) = 2x, then f -1(x) =

Answer 1

the inverse function is a function that acts opposed to the original function in such a way that it gets a value of y and gives you a value of x. However, the original function does it but in a reversed way in which upon introducing an x to its argument it will give you a value of y.

Answer 2

got it?

Answer 3

so the reverse of 2x would be -2x? right?

Answer 4

would that be the right answer for that inverse problem

Answer 5

e.g., consider the simple function f(x)=2x. let's know what f(x) would get for x=2. Substituting x=2 in 2x gives f(2)= 2*2=4. then the inverse is such that it must turn the input value y=4 into out put value x=2. We must look for a candidate that does it for us. If we choose f(x)=1/2x it would get the desired result because f(4)=1/2*4=2

Answer 6

f(x) = 2x is a rule. You give it a number and it returns a new number
example: we give it 1, and it returns 2
or we give it 10, and it returns 20

the inverse takes the *new number* example, the 20, and returns the original number e.g. the 10

Answer 7

okay that kind of makes sense

Answer 8

if you have f(x) and it returns a number y
then f^-1 will take the y and return back the original x

the inverse "undoes" the function. we start x --> function --> y --> inverse --> x

using formulas, we would write that as
\[ f^{-1}(f(x)) = x \]
which is a complicated way of saying the inverse undoes the function

Answer 9

if you have
f(x)= 2x

and we put in x=0 we get out 0. (0,0) is the (in, out) pair
if we put in 1 we get out 2, (1,2)
put in 2, we get 4 (2,4)

the inverse would swap those pairs: (0,0), (2,1), (4,2)

Answer 10

oh okay I get it now thank you

Answer 11

the way to find the inverse is use the idea that (x,y) pair for the function becomes (y,x) for the inverse.

we relabel f(x)=2x as y= 2x
swap y and x: (make y an x and vice versa) to get x= 2y
now "solve for y"
you start with
2y= x
divide both sides by 2:

2y/2 = x/2

on the left 2/2 is 1, and we get
y = x/2
or
\[ y = \frac{1}{2} x \]

Answer 12

and finally, relabel y as f^-1
\[ f^{-1}(x)= \frac{1}{2}x \]