We’re finishing up our unit on function operations. Yesterday we developed the definition of inverse functions (using only linear functions) and graphed to demonstrate the symmetry. Time to see what we have learned:

Many students’ instincts let them down on the first pair, believing them to be inverses. But after my prodding that they somehow verify the inverse relationship, we soon verified that f(g(x)) did not produce the result we desired. The second example was then complete easily.

But what about that third problem? They sure seem like inverses. One student offered his proof for the pair:

They are inverses just because I know.

Sometimes ideas in math are just that obvious, and maybe we don’t need to prove them specifically.

On the board, we “proved” that both f(g(x)) and g(f(x)) both seem to simplify as x. And a few numerical examples help show this:

- f(g(5)) = 5
- g(f(10)) = 10
- g(f( -6 )) = 6……. ruh roh……

Students in my class have not been exposed to a formal definition of the square root function, and this led to a nice discussion of absolute value, and the need to restrict the domain in order to consider inverses. Planting seeds for algebra 2, which many of my students will take next semester, is always a bonus.

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Something I did this year was start with the idea that inverse functions “undo” each other, which means “opposite operations in the opposite order”. ( That is not mathematically precise language, but conveys the idea.) I had them list, in order, the operations performed on x in the original function. Then list the inverse of each operation, but starting with the last step first, to find the inverse relation. For your #1 above, in f(x) we first multiply x by 1/2 and then add 3. So the inverse would be to subtract 3 first and then multiply by 2. It also dawned on me that we do the same thing when we solve equations. We solve for a variable in an equation by doing the order of operations in reverse, undoing each operation from the outside in. I wish that idea had dawned on me earlier.

It’s a great analogy – students can often sense when functions are inverses, and tying it to steps for solving an equation is a great idea.