Education.com

# Inverses of Circular Functions Help

(not rated)

## 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.

150 Characters allowed

### Related Questions

#### Q:

See More Questions

Welcome!