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When we talk about the domain of a function, we're referring to all the possible input values (often x-values) that the function can accept without causing any problems. Two common situations lead to domain restrictions: when we have a square root or a fraction.

For the given function, we need to avoid any values that make the inside of a square root negative or the denominator zero.

Let's break it down for the function:

• The square root in the numerator: It requires the inside to be non-negative. This means we need to solve the inequality \( z + 3 \geq 0 \) which simplifies to \( z \geq -3 \).
• The denominator: We must avoid values that make it zero. For \( z - 2 \) this happens when \( z = 2 \).

So, we gather points where issues can occur and exclude them from the domain.

Inequalities are mathematical expressions that show the relationship between two values when they aren't equal. They often include <, >, ≤, and ≥. When finding the domain of a function, inequalities can help identify where the function is defined.

For example, with the function \( h(z)=\frac{\sqrt{z+3}}{z-2} \), we need to set up and solve two inequalities:

• For the square root part, set \( z + 3 \geq 0 \). This solves to \( z \geq -3 \), meaning it can be any value higher than or equal to -3.
• For the denominator, find where \( z - 2 eq 0 \). This tells us \( z eq 2 \), meaning it cannot be exactly 2.

Combining these conditions using inequalities clearly shows the range where the function is valid.

Rational functions are ratios of two polynomial functions, such as \( \frac{P(x)}{Q(x)} \). They are common in many areas of mathematics and often present special considerations for their domains.

When dealing with rational functions, one key aspect is ensuring that the denominator is never zero because division by zero is undefined. In our exercise with \( h(z)=\frac{\sqrt{z+3}}{z-2} \), this is why we must exclude \( z = 2 \) from the domain.

Additionally, other elements like square roots in the numerator can impose further restrictions, requiring careful analysis of valid input values.

Overall, by identifying where the numerator and denominator each cause issues and combining those restrictions, we can determine the rightful domain of such rational functions.