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**Additional Polar Coordinates Practice**

**PROBLEM 1**

Provide an example of a graphical object that can be represented as a function in polar coordinates, but not in Cartesian coordinates.

**SOLUTION 1**

Recall the definitions of the terms *relation* and *function* from Chapter 6. When we talk about a function *f* , we can say that *r = f* ( *θ* ). A simple function of *θ* in polar coordinates is a *constant function* such as this:

*f* ( *θ* ) = 3

Because *f* ( *θ* ) is just another way of denoting *r* , the radius, this function tells us that *r* = 3. This is a circle with a radius of 3 units.

In Cartesian coordinates, the equation of the circle with radius of 3 units is more complicated. It looks like this:

x ^{2} + *y* ^{2} = 9

(Note that 9 = 3 ^{2} , the square of the radius.) If we let *y* be the dependent variable and *x* be the independent variable, we can rearrange the equation of the circle to get:

*y* = ±(9 − *x* ^{2} ) ^{1/2}

If we say that *y* = *g* ( *x* ) where *g* is a function of *x* in this case, we are mistaken. There are values of *x* (the independent variable) that produce two values of *y* (the dependent variable). For example, when *x* = 0, *y* = ± 3. If we want to say that *g* is a relation, that’s fine, but we cannot call it a function.

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