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To determine whether a point is on the graph of a function, we perform a process called function substitution. This means plugging the x-coordinate of the point into the function to see if the resulting y-value matches the y-coordinate of the point. Let's break it down:

Start with the given point and function. Here, the function is \( g(x)=\frac{x^2 + 4}{x + 2} \) and the point to check is \( \left(\frac{2}{3}, \frac{5}{3}\right) \).
Substitute the x-value from the point into the function:
\[ g\left(\frac{2}{3}\right) = \frac{\left(\frac{2}{3}\right)^2 + 4}{\frac{2}{3} + 2} \]
This gets us started on checking if the point is on the graph by computing if \( g\left(\frac{2}{3}\right) \) equals \( \frac{5}{3} \).

Simplifying fractions is essential in algebra to make expressions easier to understand and compare. Here’s a quick guide on how to simplify the fraction we encountered in our problem:
When you substitute \( \frac{2}{3} \) into \( g(x) \), you get an expression: \ \( g\left(\frac{2}{3}\right) = \frac{\frac{4}{9} + 4}{\frac{2}{3} + 2}\)
Convert everything to have common denominators where possible. Transform \( 4 \) into a fraction with denominator 9: \( \frac{36}{9} \).
This makes our substitution:
\ \[ g\left(\frac{2}{3}\right) = \frac{\frac{4}{9} + \frac{36}{9}}{\frac{2}{3} + \frac{6}{3}} \ = \frac{\frac{40}{9}}{\frac{8}{3}} \]
Simplifying the numerator and the denominator helps us easily compare it to the given point \( \frac{5}{3} \).
Multiplying and dividing fractions is often necessary, as seen here: \( \frac{\frac{40}{9}}{\frac{8}{3}} = \frac{40}{9} \times \frac{3}{8} \).
Finally, simplify: \ \[ \frac{120}{72} = \frac{5}{3} \]

Algebraic manipulation is a powerful tool for solving equations and simplifying expressions. Let's break down some key manipulations we used:
1. Combining like terms: Identify terms that can be added or subtracted directly. For instance, in \( \frac{\frac{4}{9} + \frac{36}{9}} \), both terms are fractions with the same denominator. So, just add the numerators: \( \frac{4 + 36}{9} = \frac{40}{9} \).

2. Changing operations: Multiplying by the reciprocal is a great way to simplify complex fractions. Here, we converted division to multiplication: \( \frac{\frac{40}{9}}{\frac{8}{3}} = \frac{40}{9} \times \frac{3}{8} \).

3. Canceling factors: When simplifying \( \frac{120}{72} \), notice that both 120 and 72 can be divided by their greatest common divisor (GCD). Dividing both by 24 simplifies the fraction: \( \frac{120 \div 24}{72 \div 24} = \frac{5}{3} \). Such simplifications are key to verifying the point \( \left(\frac{2}{3}, \frac{5}{3}\right) \) lies on the graph.

To verify if a point is on the graph of a function, follow a systematic process:
1. **Identify the problem:** You need a function and a point to verify. In our case, the function is \( g(x)=\frac{x^2 + 4}{x + 2} \) and the point is \( \left(\frac{2}{3}, \frac{5}{3}\right) \).

2. **Function substitution:** Substitute the x-coordinate of the point into the function.

3. **Calculate and simplify:** Perform any necessary calculations to simplify the result.

4. **Compare values:** Check if the calculated value matches the y-coordinate of the point exactly. If they match, as they do here: \(g\left(\frac{2}{3}\right) = \frac{5}{3}\), then the point is indeed on the graph.
This methodical approach ensures accuracy and helps us visually confirm relationships between algebraic functions and their graphical counterparts.