The uploaded image contains various mathematical problems, including functions, scientific notation, equations, and simplification problems. I'll help you solve these step by step.

Let's start with the first problem in the image:

Problem 21:

You are asked to write the rule of the function with words and as an equation, then find the missing number in the table.

The table provided is:

Step 1: Identifying the pattern

Looking at the relationship between the x-values and the y-values, we can observe the following:

• When $x = -1$, $y = -3$
• When $x = 3$, $y = 5$
• When $x = 4$, $y = 7$
• When $x = 6$, $y = 11$

The changes in y-values appear to be consistent. To confirm, we check the differences in y-values:

• $5 - (-3) = 8$
• $7 - 5 = 2$
• $11 - 7 = 4$

Thus, it seems that y increases by a constant amount based on x. From this, the rule for the function can be written as:

Step 2: Write the rule of the function

In words: "For each increase of 1 in $x$, $y$ increases by 2."

As an equation: $y = 2x + 1$

Step 3: Solve for the missing y-value when $x = 0$

Now, we use the equation $y = 2x + 1$ to find the missing value when $x = 0$: $y = 2(0) + 1 = 1$

So, the missing number in the table is 1 when $x = 0$.

Would you like me to explain any specific part further or proceed to the next problem? Let me know!

Related Questions:

1. How would the equation change if the pattern in the table changed?
2. How do you determine if a function is linear from a table of values?
3. What is the difference between a function and a relation?
4. Can you use the same method to find missing values in other types of functions?
5. How would you verify that a function is correct using different methods?

Tip: Always check the differences between y-values for a quick hint to identify linear functions!