How do I find the inverse of a function, and tell if the inverse itself is a function?

Answer 1

Put the letter y where it says f(x). So now you have an equation with x and y only. If you switch all the x's and y's, you got your inverse but now you have to get rid of the y, because it was not given to you in the original problem.

Solve for y, meaning get y alone on one side, and then replace it with f^-1(x) and there is your inverse.

The inverse is a relationship and you check its functionality just like how you check other functions for functionality. Plot it and then do the vertical line test.

In order to ensure that the inverse will be function, you have to check whether the original function is one-to-one and onto (injective and surjective also known as bijective).

For example, if I had f(x) = x^2+1, I get
y=x^2+1
x=y^2+1
y=plus minus sqrt(x-1)
f^-1(x) = plus minus sqrt(x-1)

which is obviously not a function, so one way to do it is to restrict the original f(x) to make it bijective.

I can say f(x) = x^2+1, x>0
So now the inverse will be a function
f^-1(x) = sqrt (x-1)

Answer 2

In order for a function to have an inverse, the function must be one-to-one, which is the same as passing the horizontal line test (the function cannot cross any horizontal line more than once).

You can solve for the inverse of some functions by algebra, but this method may be difficult and you just can't solve for the inverse algebraically for some functions.

Finding the inverse of a function is much easier graphically. First you graph the function y = f(x). Then you reflect the function on the line y = x. The resulting image is the inverse, whether or not it is a function.

For example, take the simple parabola y = x² opening up. Reflecting its graph on the line y = x, you get a horizontal parabola opening towards the positive x-axis. The resulting inverse is not a function, but it still acts as the inverse of the function f(x) = x².

Answer 3

when you look at the function y=f(x) you are saying for each value of x there is a specified value of y. now, by definition inverse means opposite so when it comes to functions the inverse would be x=f(y) if it exists and the rules of functions apply then the inverse is also a function
for functions of y no x values can be duplicates, so for functions of x no y values can be duplicated
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Answer 4

lety = f(x) be a function

like y = f(x) = 4x

to find the inverse, replace x with y (just change y to x and x to y)==> x = f(y)

like

x = 4y

but y must be a funtion whose variable is x; we don't want y to be a variable for x function.

so x = 4y ==> y(inverse) = x/4.

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nd i think, there is only a function whose inverse is itself which is y=x

Answer 5

If you have a function and invert it by dividing it into 1.0, the result will always be a function except for the case when the first function becomes zero.

Answer 6

if the inverse is in y = sqrt(ax + b) form, then its not a function.