Algebra 2

I'm having a lot of issues with this one.. I could really use some guidance please!

Formula: f( g(x) )
Here are the Equations:
f(x)= -2x+3
g(x)= x^2 -1
h(x)=7

I would GREATLY appreciate it if someone could assist me on this. Thank you (:

These are not equations, they are functions. The difference is very important.

A function is a recipe. In these cases you put in one number, and you get out another number. I don't do other people's homework, but I will help you get started. Take

f(x) = -2x + 3

If I put in the number 4, what is the result? (Or another way to word this is "what is the result of f(4)?").

EDIT
I think you should have written
h(x)=fg(x)
so
7=-2(xsqd-1)+3
which is a quadratic equation ...solve for x

No fresco.

The form f(g(x)) is an example of function composition. This is an important topic covered in high school algebra.

That being said, you should let her work through this. Understanding functions is a key to going further in mathematics, calculus is a study of function. If you do her homework for her you might make it more difficult for her to gain insight into this important topic.

Can we work through the problem in a couple of steps?

No max. I teach maths and was careful not to complete the homework , but to suggest what it actually was about.

Then, I think you made a pedagogical mistake that a lot of algebra teachers make. I worked for a while on a team to develop an algebra curriculum. Functions are fundamental to any mathematics past algebra, and algebra teachers have traditionally done a poor job teaching functions.

Students are taught to always "solve for x" as a knee-jerk reaction to seeing the equals sign . This is not a good thing for them to do as it gets in the way of working with functions. Functions deal with relationship between quantities, and understanding this relationship is the core of the mathematics. "Solving" for x isn't the point.

I doubt that this exercise (based on this problem) deals with solving for x. The point is to derive the function that represents the composition of these functions.

For Hayley, I would like to help you reason through the problem to develop an understanding of how function composition works.

If this would be helpful to you, please play along (if not, just ignore me).

..and I think you are making the pedagogical mistake of assuming that all high school students are going on to study advanced maths. The technique employed up to age 15 (which is the level of this problem in the UK) are those which go from 'the familiar'(equations y =...)to 'the unfamiliar' (f(x)=...), in which 'the answer' is still the main issue. Obviously mapping diagrams can be brought to extend the concept of 'function' to composite functions but this forum is hardly the place for such embellishment.

The form that she was given is

f(x)= -2x+3

This is defining a function. The problem then goes on to ask her to find a composition of two functions. Clearly she is studying the topic of functions. We can talk about pedagogy in general, and we may disagree about whether algebra students should study functions or not. But in this case, clearly the class is about functions. She should learn about functions.

There is an important difference between equations and functions. Pretending that they are the same thing will lead to a wrong understanding of mathematics.

Consider the following three expressions

2x = x + x (an identity).
8 = 2x + 2 (an equation with a solution)
f(x) = 32 + 9/5x (a useful function)

In only one of these three cases does "solving for x" make any sense. (mathematicians note that interestingly enough the "equal" sign has a different meaning in each case, some people suggest that we use different symbols for each, this may be an interesting topic if anyone would like to engage).

If our student doesn't leave this discussion understanding that these three expressions are very different, and specifically how to work with functions... she will not master the subject. Equations are not the same as functions. They aren't used the same way or treated the same way by mathematicians, scientists or engineers.

BTW. I find the thumbs on this thread amusing... math isn't a popularity contest. I would invite HayleyBoo to share this little discussion with her teacher. I suspect based on the interesting subject matter inherent in the original question that her teacher may agree with me.

I don't agree with your analysis of what her 'problem' was about, but I do agree about the 'thumbs down' nonsense.

Thank you fresco. I would much rather have an interesting discussion (or even just have you tell me I am full of crap) than these random thumbs.

Hayley should talk to her teacher. I would be very curious to know what he or she thinks.

Fresco, I am curious. Are you familiar at all with the work done by Judah Schwartz? He shaped a lot of my ideas about algebra pedagogy. I would be curious to know what you think of his ideas.

No. Not familiar with Schwartz. I might look him up. Thanks for the reference.

Fresco, I am disappointed that you wouldn't engage with me on the difference between equation and function. There has been a lot of recent research done on how kids understand functions... the ideas here are making their way into the mainstream,

Although textbook manufacturers don't like them, if you take math education classes at Harvard you will be exposed to them.

A good place to start is to look at the different ways the equal sign is used, and the inherent confusion that algebra students have because of this (which is easy to show by asking the students simple question).

When a student starts to learn computer programming (as more and more do) understanding the different ways the equal sign is used becomes even more important (and a common source of confusion).

Again Hayley should ask her teacher about this. I would bet you dollars to doughnuts that she is learning about the composition of functions. When I taught physics, I worked to make sure my students understood how functions were used in science, they were important to the subject matter that I was teaching and some of the students had some interesting misunderstandings about math.

Would anyone be interested in this discussion?

If this discussion is ending here... I am still going to tell my illustrative amusing story anyway.

When I taught high school physics, I put this problem on the first test.



A surprising number of students would do what they were taught in math class.

They would set up a ratio and "solve for x", and they would come up with an answer that fit the template they were given in math class. Of course this answer was wrong... solving for x is meaningless if you don't understand the math.

The students who gave me a smart-ass answer to this question would invariably be my best students. They thought about what they were doing to make sure it matched with their understanding of the problem. This is why there is a movement in math education to give students a deeper, truer understanding of algebra instead of just solving for x.

That appears to be just a more convoluted example of the old chestnut:

"If it takes 4 minutes to cook an egg, how long does it take for 2 eggs ?"

The point is surely that the efficacy of mathematical modelling is semantically context sensitive* rather than indicative of any pedagogical intricacies about the usage of mathematical symbols. Note that relationship between modelling and our views of 'reality' is an issue of far greater philosophical significance than the teaching of mathematics per se.
(See for example references to Lakoff and Nunez 'Where Mathematics Come From').

* The phrase 'semantically context sensitive' involves a 'social expectancy ' element in which the relative social status of questioner and respondent may be involved.

Yes Fresco!

And this is the argument that I would make about functions and equations. Functions are about modelling with "semantic context"... a function focuses on the relationship between one variable (in this case the number of eggs) and another variable (the cooking time).

Let's consider the function

cooking_time(N) = 4

This represents my hypothesis... which may or may not be true. As a scientist I would want to run an experiment that would end up with a trial with hundreds of eggs... there may be an additional term.

Notice, this function not an assignment or an identity. You can't solve for N, nor are we defining a variable called cooking_time. We are defining a function.

If after experimentation we find that each additional egg requires a bit more time, we may find the function is really

cooking_time(N) = 4 + N/100

Again, solving for N is not the point, and we are still not defining a "cooking_time" as an identity. We are defining a relationship between two quantities.

Once we set up a function to model the number of eggs to cooking time relationship we can start doing real mathematics. The context is built into the model and into the understanding of the learner.

The point I would make (and that I think many experts in math education would make) is that engineers and mathematicians use functions as a core part of how they work with mathematical ideas.

Yet in algebra we do this funny thing to students. We start them off of with equations (not functions) and get them to do this little procedural dance to solve for 'x'.

Then we throw them functions, which look an awfully lot like equations but are different in core ways. This is intellectually jolting, and high school curricula don't at all do a good job of explaining the difference. It is pretty easy to show (if you are around high school students) that this method leads to some pretty deep misunderstandings of mathematics.

Students are perfectly able to understand this level of math using functions and solution sets. The advantage is that this puts the focus of the math learning on modelling and thinking about the context.

I have seen this type of curriculum work... where functions are introduced near the beginning of Algebra 1. Equations aren't that important (once you learn to work with functions), I don't know why we insist on making them the focus of Algebra class.

In any case, in the Algebra Classes at the high school where I taught, in the middle of Algebra 2, the kids started working with functions after years of working with equations. They would come into my physics classes mathematically confused.

That is why I assume that I understand Hayley's homework. It looks awfully familiar to work that my students would be asked to do. And yes, the math teachers were trying to shift their focus into thinking about functions as "objects".

Students in Algebra 2 have to do function composition, where they can take f(g(x)) and give the answer... without ever solving for x. The point is to understand function composition.... which is a little difficult if you haven't really understood what functions are.

Yes, I'm learning about compositions of functions. Sorry I responded so late!
@maxdancona
@@fresco