**Inverses of Circular Functions Practice Problems**

**Practice 1**

Is there such a thing as a function that is its own inverse? If so, give one example.

**Solution 1**

The function *f* ( *x* ) = *x* is its own inverse, and the domain and range both happen to span the entire set of real numbers. If *f* ( *x* ) = *x* , then *f* ^{–1} (y) = *y.* To be sure that this is true, we can check to see if the function “undoes its own action,” and that this “undoing operation” works both ways. Let *f* ^{–1} be the inverse of *f* . We claim that *f* ^{–1} [ *f* ( *x* )] = *x* for all real numbers *x* , and *f* [ *f* ^{–1} ( *y* )] = *y* for all real numbers *y.* Checking:

*f* ^{–1} [ *f* ( *x* )] = *f* ^{–1} ( *x* ) = *x*

*f* [ *f* ^{–1} ( *y* )] = *f* ( *y* ) = *y*

It works! In fact, it is almost trivial. Why go through such pains to state the obvious? Well, sometimes the obvious turns out to be false, and the wise mathematician or scientist is always wary of this possibility.

**Practice 2**

Find another function that is its own inverse.

**Solution 2**

Consider *g* ( *x* ) = 1/ *x* , with the restriction that the domain and range can attain any real-number value except zero. This function is its own inverse; that is, *g ^{–1}* (

*x*) = 1/

*x*. To prove this, we must show that

*g*

^{–1}[g(

*x*)] =

*x*for all real numbers

*x*except

*x*= 0, and also that g[g

^{–1}(y)] =

*y*for all real numbers

*y*except

*y*= 0. Checking:

*g* ^{–1} [g( *x* )] = g ^{–1} (1/x) = 1/(1/x) = *x*

*g* [g ^{–1} ( *y* )] = g(1/y) = 1/(1/y) = *y*

It works! This is a little less trivial than the previous example.

**Practice 3**

Find a function for which there exists no inverse function.

**Solution 3**

Consider the function *h* ( *x* ) = 3 for all real numbers *x* . If we try to apply this in reverse, we have to set *y* = 3 in order for *h* ^{–1} ( *y* ) to mean anything. Then we end up with all the real numbers at once. Clearly, this is not a function. (Plot a graph of it and see.) Besides this, it is not evident what *h* ^{–1} ( *y* ) might be for some value of *y* other than 3.

Practice Problems for these concepts can be found at: Graphs and Inverses Practice Test

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