SOLUTION: Tutor can you assist me with this problem* I have to find the Domain for f(x)= sqrt symbol x^2-4x-21 Thank you very much****

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Question 141126: Tutor can you assist me with this problem* I have to find the Domain for f(x)= sqrt symbol x^2-4x-21
Thank you very much****
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=sqrt%28x%5E2-4x-21%29 Start with the given expression

Remember you cannot take the square root of a negative value. So that means the argument x%5E2-4x-21 must be greater than or equal to zero (i.e. the argument must be positive)

x%5E2-4x-21%3E=0 Set the inner expression greater than or equal to zero


%28x%2B3%29%28x-7%29%3E=0 Factor the left side


%28x%2B3%29%28x-7%29=0 Set the left side equal to zero


Set each individual factor equal to zero:

x%2B3=0 or x-7=0

Solve for x in each case:

x=-3 or x=7


So our critical values are x=-3 and x=7

Now set up a number line and plot the critical values on the number line

number_line%28+600%2C+-10%2C+10%2C-3%2C7%29



So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than -3 (notice how it's to the left of the leftmost endpoint):

So let's pick x=-4

x%5E2-4x-21%3E=0 Start with the given inequality


%28-4%2B3%29%28-4-7%29%3E=+0 Plug in x=-4


11%3E=+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.


So part the domain in interval notation is (]





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Let's pick a test value that is in between -3 and 7:

So let's pick x=2

x%5E2-4x-21%3E=0 Start with the given inequality


%282%2B3%29%282-7%29%3E=+0 Plug in x=2


-25%3E=+0 Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


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Let's pick a test value that is greater than 7 (notice how it's to the right of the rightmost endpoint):

So let's pick x=8

x%5E2-4x-21%3E=0 Start with the given inequality


%288%2B3%29%288-7%29%3E=+0 Plug in x=8


11%3E=+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

So part the domain in interval notation is [)





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Summary:

So the domain in interval notation is:


(] [)