Finding the inverse to a one-to-one function

Question

I need help find the inverse to this one-to-one function. The function is f(x)=2/3+x

Answer 1

1. Remember that f(x) is the same thing as y so y=2/3+x
2. swap x and y in the equation x=2/3+y
3. Solve for y , y=x-2/3
4. You are done or you can name the inverse function as f^-1(x)=x-2/3

Answer 2

Remember the definition of an inverse:

1. Graphically, the inverse of a function will be reflected over the diagonal line y=x and also be a function. Recall also that the domain of the original function becomes the range of the inverse and the range of the original function, the domain of the inverse.
2. Algebraically, by definition, when you compose a function with it's inverse the result will be 'x'. For example:

f(x) = x² - function

g(x) = √x - inverse

So, f(g(x)) = x and g(f(x)) = x

To solve for an inverse algebraically, recall that f(x) is the same as y. f(x) literally means, "the function's value (f(x) or y) is a function of (or dependent) on the value of x you choose (independent). You simply:

1. Rewrite the equation of the function with y instead of f(x)
2. Switch the x and y (switch the domain and range)
3. Solve for y to obtain the inverse

In this case:

1. y = 2/(3+x)
2. x = 2/(3+y)
3. x(3+y) = 2 (cross multiply) -> 3x + xy = 2 (distribute) -> xy = 2-3x (subtract 3x off both sides) ->

y = (2-3x)/x both sides by x. If you separate the fractions it can be written as y = (2/x) - 3