Composite Function Difficulties in Algebra 2

I taught composite functions in Algebra 2 this Tuesday and there were a few students who really struggled with understanding.

Here are two functions I might use: $f\left( x\right) =x-3$ and $g\left( x\right) =x^{2}$ .

I start with $f\left( 2\right) =2-3 =-1$ and emphasize that whatever is in the place where the 2 is in $f\left( 2\right)$ gets substituted into the x in the rest of the function. I also talk about what is in that spot is what x is equal to.

I even go so far as to say, if there is a “house” and draw a pic of one there then x is a house and would be substituted in place of the x. $f\left( house\right) =house-3$ .

Then when we move to composite functions and the problem is $f\left( g\left(2\right)\right)$ I have them calculate the inside first… $g\left( 2\right) =2^{2}=4$ and then substitute 4 in place of $g\left( 2\right)$ resulting in $f\left( g\left(2\right)\right)=f\left( 4\right)$ which then is $f\left( 4\right) =4-3=1$ .

The big jump is then to calculate $f\left( g\left( x\right) \right)$ . I do this the same way as when there is a number in it. I replace $g\left( x\right)$ with $x^{2}$ and get $f\left( x^{2}\right)$ . Then $x^{2}$ is in the x spot, so I substitute it into the x in $f\left( x\right)$ . $f\left(x^{2}\right) =x^{2}-3$ , and wahla done, simplify if you need to.

This made complete sense to me and most of my students, but there were some that even came in for help that could not grasp how to do the problems, which told me that they were not getting the function notation idea in some way.

I thought about it while I slept for the last couple of days and think I maybe have another way of going about it.

What if I calculate $f\left(g\left( x\right) \right)$ by first substituting $g\left( x\right)$ into the equation for $f\left( x\right)$ like this… $f\left( g\left( x\right) \right) =g\left( x\right) -3$ . Then say, “Hey, what is $g\left( x\right)$ and substitute it in.

$f\left( g\left( x\right) \right) =g\left( x\right) -3=x^{2}-3$

Maybe you already do it this way…

Maybe you have a better way…

Any ideas on how to teach this more efficiently so all students can grasp it?

Added 9/14/12
Ok, so I was just thinking about this and thought that I should come up with two real life functions that do something interesting to students and then do a composite of them. Maybe I’ll come up with some and do a post on it.

4 comments on “Composite Function Difficulties in Algebra 2”

crazedmummy
September 14, 2012

I use colors. This lets the students see which items come from where.

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kaleb40
September 17, 2012

That’s a great idea! I have used colors in the past but forgot about that this go around.

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Math teacher
May 6, 2015

Something that has worked well for me is including f(x+4) in the basic evaluating of functions before we get to composition. Then when we get to composition of functions, and g(x) = x + 4, they understand how to evaluate f(g(x)), aka f(x+4).

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- kaleb40
  May 8, 2015
  
  Thanks, that’s a good idea.
  
  Reply

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Composite Function Difficulties in Algebra 2

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To Accumulate a Rate — Integrate!

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