Now suppose we start with f(x)=2x³ - 5 . What does the graph look like? (Use a shift and a stretch of y = x³ to find the graph.). Note that the function is increasing over the whole real line. If we are given a number (e.g., 456) as a y-value, can we find the corresponding x value? The graph indicates that we should be able to find exactly one x so that f(x) = 456. We could approximate by tracing and zooming in to a point with the y coordinate of 456 to approximate (if necessary).

Another way is to think of the function as creating a table (possibly infinite) with the x-values in the left-hand column and the y or f(x) values in the right-hand column. (Example) Much of the time we start with a number in the left-hand column and compute and look up the value in the right-hand column. However, it is possible to start with values in the right-hand column and ask what the related value(s) are in the left-hand column. (Try this with y = x³ - 5. You might want to find x when y = 20 or when y = 1568). Suppose you are given a value c in the column for y and you want to find the x-value a which gives you f(a) = c. We can solve the equation 2a³ - 5 = c for a: a = [(c + 5)/2]^1/3.

If we never have to make a choice between two different values in the left-hand column (or on the x-axis), then this process of starting with a number on the right (on the y-axis) and finding the corresponding value on the left (on the x-axis) defines a function. If we are going to have to repeat this process many times, we especially want to find the rule for this function that gives us the value of the independent variable for f when we start with a value for the dependent variable.

Suppose f(x) = x³ - 5 and g(x) = [(x + 5)/2]^1/3 . Find f(g(x)) and g(f(x)).

What we are asking is the following then: Given a number c in the y-value column or on the y-axis can we find a single number a such that f(a) = c. If we can find this number a each time we are given a value c from the range, then we have defined a function called the inverse of f. We will say that g is the inverse of f if and only if whenever f(x) = y then g(y) = x. Of course, we could also write g is the inverse of f if and only if whenever g(x) = y then f(y) = x. If we start with one of these two expressions and use substitution in the other, then we obtain another way of proving (or defining) that two functions f and g are inverses: f(g(x)) = x for all x in the domain of g and g(f(x)) = x for all x the domain of f.

PROBLEM 1. Let f(x) = (x + 1)/(x - 3) . Find the inverse of f.
Remember g(x) = y defines the inverse of f iff f(y) = x That means we want to solve the equation x = (y+1)/(y-3) for y

                               x = (y + 1)/(y - 3)
clear of denominators          x(y - 3) = y + 1
clear of parentheses           xy - 3x = y + 1
collect y-terms on one side    xy - y = 3x + 1
factor out y                   y(x - 1) = 3x + 1
isolate the y (divide by x-1)  y = (3x + 1)/(x - 1)
This defines the inverse g     g(x) = (3x + 1)/(x - 1)

We can check by finding  f(g(x)) or  g(f(x))
                   f(g(x)) = f((3x+1)/(x-1)) 

                             (3x+1)/(x-1) + 1
                           = _________________
                             (3x+1)/(x-1) - 3

(clear of denominators)     (x-1)[(3x+1)/(x-1) + 1]
                          = ________________________
                            (x-1)[(3x+1)/(x-1) - 3]

                            3x+1 + (x-1)
                          = _____________
                            3x+1 - 3(x-1)

                          = 4x/(3x+1-3x+3)
                          = 4x/4
                          = x

Steps to finding the inverse of a function: y = f(x)

• Replace x by y in the original function (to obtain x = f(y))
• Solve x = f(y) for y. If there is a unique solution then f has an inverse. If there is a solution but it is not unique (e.g., solving x = y² yields two solutions y = x^1/2 and y = - x^1/2, then there needs to be a restriction on the original function's domain so that only one solution results (e.g., with y = x² we restrict the domain to the non-negative reals. Then the only solution possible is y = x^1/2 since y must lie in the domain of f)
• Write g(x) = solution function (or f^-1)(x)

PROBLEM 2. Let f(x) = (3x + 8)^1/2 (for x > -8/3). Find the inverse of f.
x = (3y + 8)^1/2
x² = 3y + 8
x² - 8 = 3y
y = (x² - 8)/3

This defines the inverse g(x) = (x² - 8)/3

To check that g is the inverse of f, compute f(g(x)) or g(f(x)). Either of these compositions should give us a value of x.