1. Composite Function

Hello,

I badly need help with a question..

Just to start of by defining the functions.

f: Q -> R where f(x) = 0,2cos(pi*x)-7
g: N -> Q where g(x) = 5x/2

h(x) = f(g(x)) therefore h(x)= 0,2cos(5x*pi/2)-7

a) What is the range of h?
b) What is the domain of h?
c) Is h surjective (onto) ?

I would be very glad if anyone could help me out..

2. Re: Composite Function

Let me know if you need more help.
Here I'm assuming you have learned about analyzing trig functions.

a) Hint: What is the amplitude of h(x)?
b) Hint: Are there any restrictions on x?
c) Hint: Use textbook definition of surjective. "In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x) = y. It is not required that x is unique; the function f may map one or more elements of X to the same element of Y."

3. Re: Composite Function

Originally Posted by palmstierna
Just to start of by defining the functions.

f: Q -> R where f(x) = 0,2cos(pi*x)-7
g: N -> Q where g(x) = 5x/2

h(x) = f(g(x)) therefore h(x)= 0,2cos(5x*pi/2)-7

a) What is the range of h?
b) What is the domain of h?
c) Is h surjective (onto) ?
First lets be clear;
$f(x):\mathbb{Q}\to\mathbb{R}\text{ where }f(x)=0.2\cos(\pi x)-7~\&~f(x):\mathbb{N}\to\mathbb{Q}\text{ where }g(x)=\dfrac{5x}{2}$
so $f\circ g(x)=h(x)=0.2\cos\left(\dfrac{5\pi x}{2}\right)-7$

Here is the graph of $h(x)$ use it.

4. Re: Composite Function

Hello,

thank you for clarifying. I’m quite new here!
However, I’ve looked on the graph and know the correct answers, but i don’t know how to get there algebraically/theoretical. And excuse my english..

5. Re: Composite Function

Originally Posted by palmstierna
Hello,

thank you for clarifying. I’m quite new here!
However, I’ve looked on the graph and know the correct answers, but i don’t know how to get there algebraically/theoretical. And excuse my english..
The range of the cosine function is $[-1,1]$
so $0.2[-1,1]=[-0.2,0.2]$ so that $[-0.2,0.2]-7=[-7.2,-6.8]$

It is true that a function is a surjection onto its range. However, you were asked if $h$ were a surjection without stating if the range or the finial set is the target. Which is it? Does it even say?

6. Re: Composite Function

I understand that the range of h is [-7.2, -6.8]. And thought that the domain is all real numbers.
However they say its wrong with regard to: f(x):ℚ→ℝ and g(x):ℕ→ℚ

7. Re: Composite Function

Originally Posted by palmstierna
I understand that the range of h is [-7.2, -6.8]. And thought that the domain is all real numbers.
However they say its wrong with regard to: f(x):ℚ→ℝ and g(x):ℕ→ℚ
Look at this table.
Now there is no universal agreement on $\mathbb{N}$ I and many others use $\mathbb{N}=\{0,1,2,3,\cdots\}$

In any case the domain of $h$ is $\mathbb{N}$ and the image is $\{-6.2,-7,-7.2\}$ as the table shows.

It makes no sense to say you are incorrect. Something is getting lost in translation.

8. Re: Composite Function

Let me try to translate as good as possible, the question was:

"Let us define f: ℚ→ℝ according to f(x)=15cos(πx)−7, and g: ℕ→ℚ according to g(x)=5x/2. In the exercise we are going to study the composite function h of f and g, that fulfills h(x)= f(g(x)) for every x in its domain.

a) What is the domain and the codomain of h. Your answer need to be clear and easy to follow.
b) Determine the range of h. Your answer need to be well-motivated (clue: Reflect over the domain of h and the periodicity of the trigonometric function)
c) Is h surjective (onto). Answer with a proof.

9. Re: Composite Function

Originally Posted by palmstierna
Let me try to translate as good as possible, the question was:

"Let us define f: ℚ→ℝ according to f(x)=15cos(πx)−7, and g: ℕ→ℚ according to g(x)=5x/2. In the exercise we are going to study the composite function h of f and g, that fulfills h(x)= f(g(x)) for every x in its domain.

a) What is the domain and the codomain of h. Your answer need to be clear and easy to follow.
b) Determine the range of h. Your answer need to be well-motivated (clue: Reflect over the domain of h and the periodicity of the trigonometric function)
c) Is h surjective (onto). Answer with a proof.

Given the responses so far, how would you answer these? We'll help you where you don't understand, but give it a shot first.

-Dan

10. Re: Composite Function

a) I would start by as easy as possible say that the domain is what x can be, therefore x is ℕ. [-∞, ∞]. The codomain is what h can take based on anything on x. Therefore between -36/5 and 36/5.
b) The range is what the function can take from the domain. The cosine function has the range of [-1,1], so 0.2[−1,1]=[−0.2,0.2] so that [−0.2,0.2]−7=[−7.2,−6.8].
c) I'm lost on how to find out if it's an onto function though.. Since they wanted me to do a proof I was thinking of something that I read: "Each surjection f: X→Y leads to a partition of X in k parts. If we have a partition in k parts we have k! surjection that appears from the partition."

11. Re: Composite Function

Lets look at the OP
Originally Posted by palmstierna
Just to start of by defining the functions.
f: Q -> R where f(x) = 0,2cos(pi*x)-7
g: N -> Q where g(x) = 5x/2
h(x) = f(g(x)) therefore h(x)= 0,2cos(5x*pi/2)-7
a) What is the range of h?
b) What is the domain of h?

c) Is h surjective (onto) ?
Now look at the latest post.
Originally Posted by palmstierna
a) I would start by as easy as possible say that the domain is what x can be, therefore x is ℕ. [-∞, ∞]. The codomain is what h can take based on anything on x. Therefore between -36/5 and 36/5.
b) The range is what the function can take from the domain. The cosine function has the range of [-1,1], so 0.2[−1,1]=[−0.2,0.2] so that [−0.2,0.2]−7=[−7.2,−6.8].
c) I'm lost on how to find out if it's an onto function though.. Since they wanted me to do a proof I was thinking of something that I read: "Each surjection f: X→Y leads to a partition of X in k parts. If we have a partition in k parts we have k! surjection that appears from the partition."
No one in western mathematics uses $\mathbb{N}$ to mean $(-\infty,\infty)$. Notice the $(c)~)$ not $[~]$!
b) The domain of $h$ is $\mathbb{N}$.
c) Makes no sense because the image of $h$ is $\{-7.2,-7,-6.8\}$. Surjection(?) what???

12. Re: Composite Function

Sorry, doing math on the evening is not my thing..

Well okey, the domain is N and the codomain is between -36/5 and 36/5.Therefore [-7.2, 7.2].
The range is correct.
And as earlier mentioned as you may have noticed, I’m lost on c).

13. Re: Composite Function

Originally Posted by palmstierna
Well okey, the domain is N and the codomain is between -36/5 and 36/5.Therefore [-7.2, 7.2].
The range is correct. And as earlier mentioned as you may have noticed, I’m lost on c).
In any case the domain of $h$ is $\mathbb{N}$ and the image is $\{-6.2,-7,-7.2\}$ as the table shows.