[original source unknown] |

**Introduction**

This (part of a) question is taken from a certain book sold in Singapore which did not credit the original sources, so I do not know which junior college or which year it is taken from. This is a common type of question regarding condition for the existence of composite functions, with a little twist. Many students do not feel confident solving this kind of question, which tests one’s understanding of concepts besides algebraic manipulation. I shall show you how.

Note that composite functions are written with a right-to-left convention i.e. the function on the right comes first. That means in the function gf, f is applied first, then g. This seems counter-intuitive. Think of it like this: gf(

*x*) means g(f(*x*)) by definition. Start with*x*. First we apply f. This gives f(*x*) i.e. f( ) wraps around the*x*. Next, we apply g, so we take g( ) and wrap it around f(*x*) to get g(f(*x*)). This is like putting on a shirt/blouse and then putting on a coat. As usual,

*metacogntion*and*heuristics*are very important for solving this question.

**Stage 1: Understanding the Problem**

What is this (part of a) question about?

Condition for existence composite functions, (domain) restriction of functions

What is the question asking?

Find the least value of

*k*and the value of*a*so that the function gf exists.**Stage 2: Planning the approach**

Have you solved a similar problem before? How was it solved last time?

Yes, it was solved by considering the condition for existence of gf, interpreting their meaning, and using appropriate inequalities. Use the “

*thinking forward*” heuristic: keep asking “what does this mean?” and “So what? ”.What is different this time?

There is an additional “

*x*not equal to something” type of condition.Do you think the same tactic can work?

Maybe. I can try.

**Stage 3: Executing the plan**

What does it mean for the composite function gf to exist?

It means the range of f (the first function) is contained in the domain of g.

Write:

*R*_{f}` \subseteq `*D*_{g}. Here f means the new f with the restricted domain.So what does this mean?

It means f(

*x*) is a member of the domain of gWrite: f(

*x*) ` \in ` (1, ` \oo `)\{2}. [all the numbers bigger than one, except the number 2]So what does this mean?

It means f(

*x*) > 1 and f(*x*) ` != ` 2.which means

*x*^{2}+ 2*x*> 1 and*x*^{2}+ 2*x*` != ` 2.

figure 1 – working forward |

We continue the line of reasoning using the technique of competing the square for the ‘>’ and ‘` != `’ inequalities. We need to be careful when taking square roots. Fortunately, in this situation, we know that

*x*+ 1 > 0, since*x*> -1 (*x*being in the domain of f). So we only need to consider the positive square root. We end up with*x*> -1 + ` \sqrt(2) ` and

*x*` != ` -1 + ` \sqrt(3) `

Obviously,

*a*= -1 + ` \sqrt(3) ` (the value that*x*is not supposed to be equal to).The statement

*x*> -1 + ` \sqrt(2) ` actually implies that a whole plethora of statements*x*>

*k*with

*k*= -1 + ` \sqrt(2) ` = 0.4142…

*x*>

*k*with

*k*= 0.5

*x*>

*k*with

*k*= 0.6

*x*>

*k*with

*k*= 1.1

*x*>

*k*with

*k*= 999

*x*>

*k*with

*k*= 9 999

*x*>

*k*with

*k*= 99 999

*x*>

*k*with

*k*= 1 000 000 000

… etc.

can be true. Among these the least possible

*k*is -1 + ` \sqrt(2) ` (of course!!!).**Stage 4: Evaluation**

Is it possible to check your answer? How?

We can store

*X*^{2}+ 2*X*as a function. For example, on the TI-84, we can store the formula into function variable Y_{1}. We can numerically evaluate Y_{1}(-1 + ` \sqrt(2) `) = 1 and Y_{1}(-1 + ` \sqrt(3) `) = 2. We can verify numerically that, for example, Y_{1}(0.5), Y_{1}(0.6159), Y_{1}(1.3546) , Y_{1}(99) … etc gives values greater than 1 i.e. values in the domain of g. We check through the above steps to make sure every step is correct and makes sense.**Stage 5: Reflection**

What did you learn from solving this question?

I learned to make use of the condition

*R*_{f}` \subseteq `*D*_{g}for composite function. I learned to

*work forward*by systematically refining the above statement. I remembered the “completing the square” technique learned from secondary school.

I remembered being careful when dealing with square roots in inequalities.

I learned to check my work using the calculator.

Anything else you have learned from this question? Post your comments below.

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