GRE.al.4 Find the domain of \sqrt{x^2-25}

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In summary, the domain of the function $f(x)=\sqrt{x^2-25}$ is all real numbers $x$ such that $x\le -5$ or $x\ge 5$. This is because the function value must be a real number, requiring that $x^2-25\ge 0$, which can be rewritten as $x\le -5$ or $x\ge 5$. Although complex numbers are not specified in the question, if all complex numbers are allowed as values, then the domain would be "all complex numbers".
  • #1
karush
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$\tiny{GRE.al.4}$
Find the domain of $f(x)=\sqrt{x^2-25}$
a. $[x\le-5]U[x\ge 5]$
b. x=5
c. $5 \le x$
d. $x\ne 5$
e. $\textit{all reals}$

well just by observation because of the radical I chose c.
but was wondering if imaginary numbers could be part of the domain alto it is not asked for here

https://dl.orangedox.com/QS7cBvdKw55RQUbliE
 
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  • #2
$x^2-25 \ge 0 \implies x^2 \ge 25 \implies |x| \ge 5$

also, look at the graph of $y = x^2-25$

you want to try another choice?

70B41EC0-A21A-4547-9AAC-5EBC8CFF5C73.png
 
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  • #3
oh because x can be $\pm 5$ and outside the graph so a.
 
  • #4
Since you ask about complex numbers, if all complex numbers are allowed as values for this function then its domain is "all complex numbers".

In order that this question make sense, we must be requiring that the function value, $\sqrt{x^2- 25}$ be a real number. That means that $x^2- 25\ge 0$.

$x^2- 25= (x- 5)(x+ 5)\ge 0$.
The product of two numbers is positive if and only if the two numbers have the same sign- both postive or both negative.

x- 5 and x+ 5 are both positive, x- 5> 0 and x+ 5> 0 so x>5 and x> -5. Of course, if x> 5 then x is also greater than -5 so $x\ge 5$ is a solution.

x- 5 and x+ 5 are both negative, x- 5< 0 and x+ 5< 0 so x< 5 AND x< -5. Of course, if x< -5 then x is also less than 5 so $x\le -5$ is a solution.

The solution set is $\{x| x\le -5 or x\ge 5\}$.
 

FAQ: GRE.al.4 Find the domain of \sqrt{x^2-25}

1. What is the purpose of finding the domain of a function?

Finding the domain of a function allows us to determine the set of values for which the function is defined. It helps us understand the behavior of the function and identify any potential restrictions or limitations.

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

To find the domain of a square root function, we need to consider the expression inside the square root. In this case, x^2-25 is the expression inside the square root. We need to find the values of x that make this expression non-negative, as the square root of a negative number is undefined. Therefore, the domain for this function would be all real numbers except for x = 5 and x = -5.

3. Can the domain of a function change?

Yes, the domain of a function can change depending on the type of function and any restrictions or limitations. In the case of the square root function, the domain can change if the expression inside the square root changes.

4. What happens if the domain of a function is not specified?

If the domain of a function is not specified, it is assumed to be all real numbers for which the function is defined. However, it is always important to consider any potential restrictions or limitations in the domain.

5. How can I graph a function with a restricted domain?

To graph a function with a restricted domain, we can simply plot the points that fall within the specified domain. For example, with the function \sqrt{x^2-25}, we would plot all points on the graph except for x = 5 and x = -5.

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