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SosoTrey · Feb 9, 2011, 06:59 PM

Hi I'm having problems with my math homework its suppose to be easy but I just don't remember how to do this which to find the domain of function such as :
f(x)=√(x+19 )
and
f(x)=1⁄ √(12x+10 )
thank you!

jcaron2 · Feb 9, 2011, 08:21 PM

The domain is the set of possible x-values that you could use which give a valid result. Most mathematical operations don't have any limitations whatsoever (for example, addition, subtraction, and multiplication work for absolutely ANY number, so those operations don't restrict the domain at all). Some operations, however, DO have restrictions. Your problems illustrate two different operations with restricted domains. The first is the square root function. In order for the result to be real, the terms inside the square root have to be greater than or equal to zero. That puts restrictions on what the value of x can be. In your second equation, you also have a division operation. That has the additional restriction that whatever's in the denominator can't be zero (division by zero is illegal).

So for your first equation, you can find the domain by solving the inequality

x + 19 >= 0

Can you tell me the inequality for the second equation?

SosoTrey · Feb 9, 2011, 08:24 PM

12x+10 >0 or 12x+10=0
is it right?

jcaron2 · Feb 9, 2011, 09:58 PM

12x+10>0 is right, which means

12x > -10
x > -5/6

Good job.

The second part you wrote, 12x + 10 = 0, is NOT right, since that would cause the denominator to go to zero. That's why you were correct to use >, rather than >= (meaning greater than or equal to) in your answer.

SosoTrey · Feb 10, 2011, 11:11 AM

Ahh OK so the domain is: (-infinity, -5/6) U (5/6, infinity)
And if I'm still right the second one is ; (-infinity, 4.5)

Unknown008 · Feb 10, 2011, 11:21 AM

No, you cannot have x as a negative number, unless you're looking for imaginary parts :rolleyes:

The range of the first function is simply:

$\text{Range} = \lbrace{f(x)\in\mathbb{R}\ :\ x\ >\ -\frac56\rbrace}$

or

$(-\frac56 ,\ \infty]$

or simply

$x\ > \ -\frac56$

This is for the second one. The first one is:

$\text{Range} = \lbrace{f(x)\in\mathbb{R}\ :\ x\ \geq\ -19\rbrace}$

or

$[-\frac19 ,\ \infty]$

or simply:

$x\ \geq \ -19$

jcaron2 · Feb 10, 2011, 11:23 AM

No, I think you're making it more complicated than it needs to be. :) it's just x > -5/6. That means the domain is (-5/6, infinity]. That's it. Period.

The other one, x + 19 >= 0, is simplified to x >= -19. So it's domain is [-19, infinity]. Notice this time there's a square bracket on the left side, indicating greater than or equal to.

Does that make sense?