A function has an inverse if and only if it is a one-to-one function. That is, for every element of the range there is exactly one corresponding element in the domain. To use an example f(x), f(x) is one-to-one if and only if for every value of f(x) there is exactly one value of x that gives that value.
f(x) = x2 is not one-to-one. x= -2 and x=2 both give f(x)=4
We can make a function one-to-one by restricting it's domain. In the previous example if we say f(x)=x2 where x>0 then f(x) is one-to-one.
The function g(x) = square root (x) is the inverse of f(x)=x2 where x>0.
Algebraic functions involve only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.
If an algebraic function is one-to-one, or is with a restricted domain, you can find the inverse using these steps.
Example: f(x) = (x-2)/(2x) This function is one-to-one.
Step 1: Let y = f(x)
Step 2: solve for x in terms of y
2xy=x-2 multiply both sides by 2x
2xy-x=-2 subtract x from both sides
x(2y-1)=-2 factor out x from left side
x=-2/(2y-1) divide both sides by (2y-1)
Step 3: switch the x and y
Step 4: Let y=f-1(x)
Some non-algebraic functions have inverses that are defined.
sin and arcsine (the domain of sin is restricted)
other trig functions e.g. cosine, tangent, cotangent (again the domains must be restricted.)
If f(x) = 10x then f-1(x) = log(x)
If f(x) = ex then f-1(x) = ln(x)