How to Find the Domain and Range of √(H^2 + 12756H)?

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To find the domain and range of the function y = √(H^2 + 12756H), it is crucial to determine when the expression inside the square root is non-negative. The inequality H(H + 12756) < 0 indicates that the product of H and (H + 12756) must be negative, which occurs when H is between the roots of the equation H^2 + 12756H = 0. The roots are H = 0 and H = -12756, leading to the conclusion that the domain of the function is H in the interval (-12756, 0). The range of the function is then all non-negative real numbers, as the square root yields non-negative outputs. Understanding these algebraic conditions allows for a clear determination of the function's domain and range.
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Homework Statement



I have the function H^2 + 12756H and I want to find the domain and range of it's square root algebraically.

Homework Equations

The Attempt at a Solution



I understand y= √(H^2 + 12756H) is undefined if H^2 + 12756H < 0, however I don't get how to find its domain and range using algebra. Maybe someone can help me.
 
Last edited:
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zaddyzad said:

Homework Statement



I have the function H^2 + 12756H and I want to find the domain and range of it's square root algebraically.

Homework Equations



The Attempt at a Solution



I understand y= √(H^2 + 12756H) is undefined if H^2 + 12756H < 0, however I don't get how to find its domain and range using algebra. Maybe someone can help me.
Can you solve for H2 + 12756H = 0 ?

If you graph y = H2 + 12756H you get a continuous graph. H2 + 12756H cannot change sign without going through zero.

All of that should help you.
 
I understand how to find the domain and range of the root function by looking at the original graph, however my question is can it be done algebraically, if so how?

And no I don't believe you can solve for H with 0 = h^2 + 12756h algebraically only with quadratic formula. The point of this post is to find out the easiest way to solve this sort of thing for domain, and would it be finding x-intercepts and deriving your domain of the root from that ?
 
Last edited:
zaddyzad said:
I understand how to find the domain and range of the root function by looking at the original graph, however my question is can it be done algebraically, if so how?

And no I don't believe you can solve for H with 0 = h^2 + 12756h algebraically only with quadratic formula. The point of this post is to find out the easiest way to solve this sort of thing for domain, and would it be finding x-intercepts and deriving your domain of the root from that ?
How purely algebraic do you want the solution to be?

Yes, it can be done!
 
zaddyzad said:

Homework Statement



I have the function H^2 + 12756H and I want to find the domain and range of it's square root algebraically.


Homework Equations




The Attempt at a Solution



I understand y= √(H^2 + 12756H) is undefined if H^2 + 12756H < 0, however I don't get how to find its domain and range using algebra. Maybe someone can help me.
H^2+ 12756H= H(H+ 12756)< 0 and the product of two numbers is negative if and only if the numbers are of opposite sign.
So either H< 0 and H+ 12756> or H> 0 and H+ 12756< 0.
(Only one of those is actually possible.)
 

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