91Ó°ÊÓ

Let's start with understanding what domain means in algebra. The domain of a function is the set of all possible inputs (x-values) for which the function is defined. For the function \ g(x) = 2 + \sqrt{3x-5} \, the expression inside the square root, \ 3x-5 \, must be non-negative.
This leads us to an important step: setting up and solving inequalities to find the domain.

For this function, we need to find when \ 3x-5 \ is greater than or equal to zero. This is because the square root of a negative number is not a real number. So, we set up the inequality:\[ 3x - 5 \geq 0 \].

Next, we solve the inequality. First, add 5 to both sides:\[ 3x \geq 5 \]. Then, divide both sides by 3: \[ x \geq \frac{5}{3} \]. Thus, the domain of our function \ g(x) \ is \[ \left[ \frac{5}{3}, \infty \right) \]
.

Solving inequalities is an essential skill in algebra. In many cases, like our previous example, it helps to determine the domain of a function. To solve an inequality, follow these steps:

• 1. First, simplify both sides if needed.
• 2. Use basic algebraic operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember, if you multiply or divide both sides by a negative number, you must flip the inequality sign.
• 3. Write down the solution set, which describes all the values the variable can take.

Let's solve the inequality from our problem step-by-step: \[ 3x - 5 \geq 0 \].

1. Add 5 to both sides: \[3x \geq 5\]
2. Divide by 3: \[x \geq \frac{5}{3} \]

The solution is \[ x \geq \frac{5}{3} \], meaning any number greater than or equal to \[ \frac{5}{3} \] is part of the solution set. This indicates the domain of our function is all x-values starting from \[ \frac{5}{3} \] and going to infinity.

Graphing functions is a powerful way to visualize how they behave. It helps confirm the domain and provides insight into the range. To graph the function \( g(x)=2+\sqrt{3x-5} \), follow these steps:

• 1. Set up your graphing calculator by inputting the function.
• 2. Adjust the viewing window to see where the function starts (from the domain) and how it behaves as x increases.
• 3. Observe the graph: determine the range by seeing the minimum and maximum y-values.

For our function, the domain \( \left[ \frac{5}{3}, \infty \right) \) means the graph starts at \( x = \frac{5}{3} \). Plugging \( x = \frac{5}{3} \) into \( g(x) \) gives us \( g\left( \frac{5}{3} \right) = 2 + \sqrt{3 \frac{5}{3} - 5} = 2 \). This is the starting point of the graph.

The range of the function is then \( \left[ 2, \infty \right) \). The square root function increases infinitely, so adding 2 means our function starts at 2 and goes upwards forever.

Using a graphing calculator or software confirms this. By graphing `g(x)=2+\sqrt{3x-5}`, you can visually verify the domain and estimate the range. This visual check solidifies your understanding and ensures your algebraic solution is correct.