How Can You Symbolically Solve for x in Mathematical Functions?

[Ryanfjdskljafl]

Hi everyone, first time on the forum, so please feel free to give me unsolicited advice on if there's a better place to post this:

if i have a function f(x), is there a way to symbolize 'solve for x'.

for example, if f(x) == 3x^2, and I'm setting f(x) = 3, I want to be able to symbolically do something like x = f_inverse(3), so that we solve for x.
3x^2 = 3, x = 1.

is there a method to represent this symbolically like what I tried doing with f_inverse above?

 

[Mark44]

I moved your thread out of the Homework section as it seemed a better fit in the General Math section.

if i have a function f(x), is there a way to symbolize 'solve for x'.

for example, if f(x) == 3x^2, and I'm setting f(x) = 3, I want to be able to symbolically do something like x = f_inverse(3), so that we solve for x.
3x^2 = 3, x = 1.

is there a method to represent this symbolically like what I tried doing with f_inverse above?

In general, no, there's not a way to solve for x in the equation y = f(x), unless the function is invertible. The condition for invertibility is that the function is one-to-one.

Your example function is not invertible. The function f(x) = 3x² is not one-to-one, so doesn't have an inverse. If you solve the equation 3 = 3x², you get two solutions: x = 1 or x = -1.

A function that is invertible is y = g(x) = e^x. This is a one-to-one function, so there is an inverse that is itself a function; namely, x = g^-1 (y) = ln(y).

Note to Fredrik, in case he steps in. I have used the notation sloppily. f above maps x to 3x². I am aware of the difference between a function f and a function value f(x).