Notes for Introductory Calculus (Math 120)

In part (c) of Example A of the previous section, we knew that the value of the function \(m(t)\) (mass of Strontium-90 remaining) was 5 mg, and wanted to know the corresponding value of its argument \(t\) (the time).

More generally it would be useful to have a formula giving time \(t\) as a function of mass \(m\text{,}\) \(t=g(m)\text{.}\) A function like this that takes values “backwards” compared to the function is the inverse of that function.

For example, with \(y=f(x)=x^{3}\text{,}\) we get back from a \(y\) value to the corresponding \(x\) value by treating \(y\) as known and solving \(y=x^3\) for the unknown \(x\text{:}\) this gives \(x=\sqrt[3]{y}\text{.}\) Thus the cube root function is the inverse of the cube function, and we write \(x=f^{-1}(y)=\sqrt[3]{y}\text{.}\)

It is often convenient to go back to using the name \(x\) for the argument of this new function too, writing \(f^{-1}(x)=\sqrt[3]{x}\text{.}\)

Graphically, one gets the graph of the inverse function \(x=f^{-1}(y)\) by flipping the graph of \(y=f(x)\) along the diagonal line \(y=x\text{.}\) Since the role of the \(x\) and \(y\) values are swapped, the domain of the inverse is the range of the original function, and vice versa.

However, this flipping does not always give the graph of a function. The graph of any function must pass the vertical line test that no vertical line intersects it more than once, and for the flipped graph to pass, the original graph must have no horizontal line intersects it more than once.

This is the Horizontal Line Test [HLT], and is exactly what is needed for a function to have an inverse. Algebraically, this means that no two different arguments \(x_1\) and \(x_2\) give the same value of the function:

Definition 1.2.1.

Function \(f\) is one-to-one if for any \(x_1 \neq x_2\text{,}\) \(f(x_1) \neq f(x_2)\text{.}\)

How do we reconcile \(f(x)=x^2\) not being one-to-one with it having an inverse, the square root function?

Be careful: we are talking about two different functions here, even though they are described using the same formula \(y=x^2\text{!}\) When we use the formula with domain all the real numbers, it is not one-to-one, and has no inverse, but when we change the domain to non-negative real numbers, that is a different function, with inverse \(\sqrt{x}\text{.}\) The difference with this smaller domain is that the graph is only the right half of a parabola, which is increasing and so satisfies the HLT.

Algebraically, for any given value \(y\text{,}\) the equation \(x^2=y\) has only one non-negative solution \(x\text{.}\)