91Ó°ÊÓ

When we talk about the domain of a function, we are discussing the possible inputs, or values of 'r', that will yield valid outputs. For the function \( F(r) = \sqrt{r+4} \), we must identify for which values of 'r' the function can be calculated without issues. Functions involving a square root only accept expressions inside them that are greater than or equal to zero because you cannot take the square root of a negative number in the realm of real numbers.
To ensure that \( F(r) \) is defined, the expression under the square root must be non-negative, leading to the inequality \( r + 4 \geq 0 \). Solving this inequality is simple: subtract 4 from both sides to find that \( r \geq -4 \). This tells us that 'r' can be any number starting from \(-4\) up to infinity.

• The domain of the function \( F(r) = \sqrt{r+4} \) is therefore \([-4, \infty)\).

Inequalities often play a crucial role in determining where a function is defined, especially when dealing with square roots or logarithmic functions. For \( F(r) = \sqrt{r+4} \), the inequality \( r + 4 \geq 0 \) is vital.
This inequality ensures that the function only takes input values that do not lead to having to calculate the square root of a negative number, which is undefined in real numbers.
Here are the basic steps to solve such inequalities:

• Move all terms involving the variable to one side so that the inequality involves \( r \) on one side and a constant on the other, like \( r \geq -4 \).
• Determine the direction of the inequality. In this case, it's "greater than or equal to," meaning \( r \) can start at \(-4\) and continue to increase to positive infinity.

These steps help to find out the set of values where the function is nicely behaved and can be evaluated without issues.

The function range is about figuring out what possible output values or 'y-values' a function can produce. For \( F(r) = \sqrt{r+4} \), since \( r \geq -4 \), the expression \( \sqrt{r + 4} \) will always be non-negative. The reason is that the smallest square root value occurs when its argument, \( r + 4 \), is zero. This happens when \( r = -4 \), so \( F(r) = \sqrt{0} = 0 \).
As 'r' increases beyond \(-4\), the value of \( r + 4 \) increases, and thus \( F(r) = \sqrt{r+4} \) gets larger as well. Therefore, all non-negative values from 0 upwards are possible outputs of the function.

• The range of the function \( F(r) \) is hence \([0, \infty)\), signifying that the smallest y-value it can output is 0, with no upper limit to the positive numbers it can reach.

This understanding helps interpret the behavior of the function's outputs based on varied inputs.