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The **domain of a function** refers to the set of all possible input values (often called 'x-values') for which the function is defined. In other words, the domain is the collection of x-values that you can substitute into the function without causing any mathematical problems. For instance, in the function given, \( g(x) = \sqrt{x^2 - 25} \), the expression inside the square root \(x^2 - 25\) must be non-negative because the square root of a negative number is not a real number.
To find the domain, set up the inequality: \( x^2 - 25 \geq 0 \). Solving this, we get:
\( x^2 \geq 25 \)
This implies:
\( x \geq 5 \) or \( x \leq -5 \)
So, the domain of \( g(x) \) is all x-values where x is either greater than or equal to 5, or less than or equal to -5.

The **square root function** involves any mathematical function that contains a square root operation. This means the function must have an expression under the square root (or radical sign). In our example, the function is \( g(x) = \sqrt{x^2 - 25} \). Remember that the square root function only produces real numbers if the expression inside the square root is zero or positive.
For example:
If \( x = 6 \), then we substitute in our function: \( g(6) = \sqrt{6^2 - 25} = \sqrt{36 - 25} = \sqrt{11} \). Similarly, if \( x = -6 \), we get: \( g(-6) = \sqrt{(-6)^2 - 25} = \sqrt{36 - 25} = \sqrt{11} \). Note that inside the square root, the negative input is squared, making it positive.

When **solving inequalities**, we aim to find the range of x-values that satisfy the inequality condition. In the given function, we need to ensure the expression inside the square root is non-negative:
\( x^2 - 25 \geq 0 \)
Start by solving for \( x^2 \geq 25 \). This can be broken down into two separate inequalities:
\( x \geq 5 \)
or
\( x \leq -5 \)
This tells us that x must be either greater than or equal to 5, or less than or equal to -5, ensuring that the expression under the square root is always zero or positive.

In **function substitution**, you replace the variable (often 'x') in the function rule with a specific value to determine the output. Let's apply this to our given function. To find \( g(6) \), we replace 'x' with 6:
Substitute 6 in \( g(x) = \sqrt{x^2 - 25} \): \( g(6) = \sqrt{6^2 - 25} = \sqrt{36 - 25} = \sqrt{11} \).
Similarly, when evaluating \( g(13) \), substitute 13 into the function:
\( g(13) = \sqrt{13^2 - 25} = \sqrt{169 - 25} = \sqrt{144} = 12 \). This substitution process allows us to compute the output values of the function effectively.
Note: If the specific value is not within the domain of the function, the function value does not exist, as in the case of \( g(3) \).