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When we talk about the domain of a function, we are dealing with the set of all possible input values, usually denoted as 'x', for which the function is defined. Essentially, it's like asking, "For which 'x' values can we safely compute a function like g(x)?"

To determine this, look out for operations within the function that have restrictions. For instance, division by zero is undefined in mathematics. So, if a function has a denominator, any value of 'x' making this denominator zero must be excluded from the domain. Similarly, for functions involving even roots or logarithms, ensure those inner values keep the operation valid.

• Check denominators and set them not equal to zero.
• Ensure values under even roots are non-negative.
• Logarithms should be greater than zero inside.

By eliminating these problematic cases, you find the domain of the function.

The absolute value function, denoted by the 'pipes' | | around a number or expression, represents the non-negative value of that number or expression. It's like asking, "How far is this number from zero?" without caring about its direction. For instance, both |3| and |-3| equal 3.

This concept modifies how we solve mathematical equations, especially when these seem embedded in a function's denominator or similar constraining factors. When you encounter expressions involving absolute values, remember some key tips:

• |a| = a if a is positive or zero.
• |a| = -a if a is negative.

So in context, for expressions like |x²-4|, it changes how the function behaves around critical points (in this case x = -2 and x = 2) as it flips any negative results to positive, impacting where the function remains defined or undefined.

Rational functions are in the form of one polynomial divided by another. The general form is f(x) = P(x)/Q(x), where both P(x) and Q(x) are polynomials. These functions are a big part of calculus, given their diverse behaviors and potential singularities where they become undefined when Q(x) = 0.

Important characteristics to consider:

• Critical points, where rational functions potentially become undefined due to a zero denominator.
• Vertical asymptotes, which occur at these critical points, signaling dramatic changes in function values close to these points.
• Horizontal asymptotes or oblique asymptotes, depending on the degrees of P(x) and Q(x), informing about behavior at extreme x-values.

When dealing with a function like g(x) = 1/|x²-4|, be sure to consider how the absolute value impacts the denominator, changing which x values are critical and if they remain undefined as seen here with x = -2 and x = 2 remaining critical as |x²-4| could still be zero.