Find the Composite Function of p & q: Relationship & Value of pq(39.72)

[Natasha1]

Homework Statement

p(x)=(2−x) / (3 + x) and q(x)= (2−3x) / (1+ x)

a) Find the function pq(x).
b) Hence describe the relationship between the functions p and q.
c) Hence write down the exact value of pq(39.72).

2. The attempt at a solution

a) I got pq(x) = x by substituting q(x) into p(x). Is this correct answer?

b) If pq(x) = x then the functions p and q are positive. Is this the correct answer?

c) pq(39.72) = 39.72 . Is this the correct answer?

Please let me know what I am doing wrong. Thank you.

 

[haruspex]

I got pq(x) = x

Yes.

then the functions p and q are positive

The question asks for the relationship between the functions.

 

[Natasha1]

Would you said they are inverse to each other?

 

[haruspex]

Would you said they are inverse to each other?

Yes.

 

[Natasha1]

They cancel each other out, right?
What about this question

Hence write down the exact value of pq(39.72)? Is that correct?

 

[haruspex]

They cancel each other out, right?
What about this question

Hence write down the exact value of pq(39.72)? Is that correct?

Yes.

 

[Natasha1]

Haruspex, could you please explain why these two functions are the inverse of one and the other?

 

[PeroK]

Haruspex, could you please explain why these two functions are the inverse of one and the other?

Assuming @haruspex has gone to bed:

If you have a function $q$ and you find another function $p$ such that $p(q(x)) = x$, then by definition $p$ is the inverse of $q$. By showing that $p(q(x)) = x$, you have shown that these are inverses of each other. In that sense, there is nothing more to say.

But, you might ask how would you find $p$, the inverse function?

If you let $y = q(x)$, then this gives you the graph of $q(x)$. To find the inverse function, you need to find $x = p(y)$. Graphically, therefore, you simply swap the x and y axes and you have the graph of the inverse function.

You can also find the function algebraically. In this example, you have:

$y = q(x) = \frac{2- 3x}{1+x}$

You need to rearrange that and solve for $x$ in terms of $y$. This gives you:

$x = \frac{2-y}{3+y} = p(y)$

And, that is your inverse function.

 

[Natasha1]

Thanks PeroK, much appreciated...

PS: Good night, haruspex :)

 

[Mark44]

@Natasha1, it's good that your title included the word "composite," because what you posted didn't look at all like the composition of two functions.

a) Find the function pq(x).

This looks like the product of functions p and q. To talk about the composition of these functions, you should write the above as $p(q(x))$, as PeroK did in post #8. Different notation with the same meaning is $(p \circ q)(x)$.