Inverses of Circular Functions Help

Introduction to Inverses of Circular Functions

Each of the circular functions has an inverse: a function that “undoes” whatever the original function does. Defining and working with inverse functions can be tricky.

What Is An Inverse Function?

What is meant by the term inverse function, or the inverse of a function? In general terms, the inverse of a function, if it exists, does exactly the reverse of what the function does. We’ll get more formal in a moment. But first, we must clarify something about notation.

When a function f has an inverse, it is customary to denote it by a superscript, so it reads f ^–1 . This superscript is not an exponent. The function f ^–1 is not the same thing as the reciprocal of f . If you see f ^–1 ( q ) written somewhere, it means the inverse function of f applied to the variable q. It does not mean 1/[f(q)]!

Here is the formal definition. Suppose we have a function f . The inverse of f , call it f ^–1 , is a function such that f ^–1 [ f ( x )] = x for all x in the domain of f , and f [ f ^–1 (y)] = y for all y in the range of f . The function f ^–1 “undoes” what f does, and the function f “undoes” what f ^–1 does. If we apply a function to some value of a variable x and then apply the function’s inverse to that, we get x back. If we apply the inverse of a function to some value of a variable y and then apply the original function to that, we get y back.

Not every function has an inverse without some restriction on the domain and/or the range. Sometimes a function f has an inverse f ^–1 without any restrictions; that is, we can simply turn f “inside-out” and get its inverse without worrying about whether this will work for all the values in the domain and range of f . But often, it is necessary to put restrictions on a function in order to be able to define an inverse. Let’s look at an example.

Square Vs Square Root

Figure 3-8 is a graph of a simple function, f ( x ) = x ² . In this graph, the values of f ( x ) are plotted on the y axis, so we are graphing the equation y = x ² . This has a shape familiar to anyone who has taken first-year algebra. It is a parabola opening upward, with its vertex at the origin.

Fig. 3-8. The relation y = x ² is a function of x.

What do you suppose is the inverse function of f ? You might be tempted to say “The square root.” If you say that, you’re right—partly. Try graphing the parabola with the x and y variables interchanged. You’ll plot the curve for the equation x = y ² in that case, and you’ll get Fig. 3-9. This is a parabola with exactly the same shape as the one for the equation y = x ² , but because the x and y axes are switched, the parabola is turned on its side. This is a perfectly good mathematical relation, and it also happens to be a function that maps values of y to values of x. But it is not a function that maps values of x to values of y. If we call this relation, g ( x ) = ± x ^1/2 , a function, we are mistaken. We end up with some values of x for which g has no y value (that is okay), and some values of x for which g has two y values (that is not okay). This is easy to see from Fig. 3-9.

Fig. 3-9. The relation y = ± x ^1/2 , while the inverse of the function graphed in Fig. 3-8 , is not a function.

What can we do to make g into a legitimate function? We can require that the y values not be negative, and we have a function. Alternatively, we can require that the y values not be positive, and again we have a function. Figure 3-10 shows the graph of y = x ^1/2 , with the restriction that y ≥ 0. There exists no abscissa ( x value) that has more than one ordinate ( y value).

Fig. 3-10. The relation y = x ^1/2 is a function if we require that y be non-negative.

If you are confused by this, go back to The Cartesian Plane Help and review the distinction between a relation and a function.