Finding the domain of a composite function help.

In summary, the composite function √2x^2+5 has a domain of all real numbers, similar to other square functions like √2-x. This is because the square of any real number is always greater than or equal to 0, and since 2x^2 is always positive, adding 5 does not change this. Therefore, the domain is not affected by adding a constant to the expression, and the domain remains all real numbers.
  • #1
nukeman
655
0

Homework Statement



Ok, I just worked out a composite function, and it left me with:

√2x^2+5)

Now, how do I find the domain from that? I don't understand that my text says the domain of that is just all Real numbers ?

What makes this different than other square functions that we are able to find the domain like: √2-x ?

Homework Equations





The Attempt at a Solution

 
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  • #2
This isn't any different from those. How would you go about finding the domain of the example function you posted?
 
  • #3
Then why did my book say the domain is all REAL numbers?

I would go:

2mg5xdj.jpg
 
  • #4
You did a bunch of work that doesn't get you anywhere.

The square of any real number is ≥ 0.

IOW, x2 ≥ 0, for all real x
so 2x2 ≥ 0, for all real x
so 2x2 + 5 ≥ 5, for all real x
 
  • #5
Kinda lost here.

So, 2x^2 + 5

the 2x^2 part is always the same, as 2x^2, x can be any real number. And why is that?

how did you get 2x^2 + 5 <= 5?

What would the domain be?

Thanks Mark!
 
  • #6
x2 is always positive, do you expect 2x2 or 2x2+5 to result a negative number?
nukeman said:
Kinda lost here.

So, 2x^2 + 5

the 2x^2 part is always the same, as 2x^2, x can be any real number. And why is that?
x can be any real number because square of every real number is defined.
 
  • #7
Oh...ok

So, everytime I get a 2x^2 in a square root, and have to find the domain, its just all real numbers? Then I just worry about what after it, like the +5?

But then, wouldn't 2x^2 + 5 be x >= -5?
 
  • #8
nukeman said:
Kinda lost here.

So, 2x^2 + 5
This is not a complete statement. So 2x2 + 5 is what?
nukeman said:
the 2x^2 part is always the same, as 2x^2, x can be any real number. And why is that?

how did you get 2x^2 + 5 <= 5?
I didn't. I got 2x2 + 5 ≥ 5
nukeman said:
What would the domain be?

Thanks Mark!

nukeman said:
Oh...ok

So, everytime I get a 2x^2 in a square root, and have to find the domain, its just all real numbers? Then I just worry about what after it, like the +5?
If you have an expression like x2, or 2x2 or .1x2, it is always guaranteed to be ≥ 0, because the square of any real number can't be negative.
nukeman said:
But then, wouldn't 2x^2 + 5 be x >= -5?
NO!

2x2 + 5 is an expression that has a value for each value of x.

x ≥ -5,
2x2 + 5 = 0,
2x2 + 5 < 0,
and
2x2 + 5 ≥ 0
are all statements that are either true or false, possibly depending on the value of x, but possibly not.

You can't compare 2x2 + 5 (an expression) and x ≥ -5 (a statement).

If you start with this statement:
2x2 + 5 ≥ 0,

you can add -5 to both sides to get
2x2 ≥ -5

This is legitimate, but unwise. Since x2 ≥ 0 for all real x, then 2x2 ≥ 0 for all x, so 2x2 is automatically ≥ -5.
 
  • #9
Oh ok, I see what you mean here. Thanks Mark.

So, for that expression, if they ask for the domain of that expression, what would it be? That is where I am lost.
 
  • #10
By that expression, I assume you mean ## \sqrt{2x^2 + 5}##, right? I don't see how you can still be asking this. You know the answer, and this thread has explained what's going on in considerable detail.
 

1. What is a composite function?

A composite function is a mathematical concept where two or more functions are combined to form a new function. The output of one function becomes the input of the next function, and so on.

2. How do I find the domain of a composite function?

To find the domain of a composite function, you first need to find the domain of each individual function. Then, you need to determine the values that are common in the domains of all the functions. These common values will be the domain of the composite function.

3. Can the domain of a composite function be different from the domains of its individual functions?

Yes, the domain of a composite function can be different from the domains of its individual functions. This is because the composite function is a new function that may have restrictions or limitations that are not present in the individual functions.

4. What are some common mistakes when finding the domain of a composite function?

Some common mistakes when finding the domain of a composite function include forgetting to consider the domain of each individual function, not accounting for any restrictions in the composite function, and not checking for common values in the domains of the individual functions.

5. Are there any shortcuts for finding the domain of a composite function?

There are no shortcuts for finding the domain of a composite function. It requires careful analysis and consideration of the domains of each individual function. However, with practice, you can become more efficient at identifying common values and restrictions in the composite function.

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