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### 2.1 Write the following equations as functions.

#### 2.1.1 \( 4y - 6x = -2 \)

To write this as a function of \(x\), we need to solve for \(y\).

Starting with the equation:
\[
4y - 6x = -2
\]
Add \(6x\) to both sides:
\[
4y = 6x - 2
\]
Now, divide the entire equation by 4 to solve for \(y\):
\[
y = \frac{6x - 2}{4} = \frac{6x}{4} - \frac{2}{4} = \frac{3x}{2} - \frac{1}{2}
\]

Thus, the function can be written as:
\[
y(x) = \frac{3x}{2} - \frac{1}{2}
\]

#### 2.1.2 \( P = \frac{(Q)}{E + 1} \)

This equation is already given as a function of \(E\). The function can be written as:
\[
P(E) = \frac{Q}{E + 1}
\]

Here, \(P\) is expressed in terms of the variable \(E\).

### 2.2 Write functions of the following table.

We are provided with a table for \(f(x)\) and \(g(x)\) as functions of \(x\).

#### 2.2.2 \(f(x)\):

The values of \(f(x)\) are:
\[
\text{For } x = 1, f(x) = 4
\]
\[
\text{For } x = 2, f(x) = 6
\]
\[
\text{For } x = 3, f(x) = 8
\]
\[
\text{For } x = 4, f(x) = 10
\]
\[
\text{For } x = 5, f(x) = 12
\]

By observing the pattern, we see that \(f(x)\) is increasing by 2 for each increase in \(x\). This suggests a linear relationship.

The function can be written as:
\[
f(x) = 2x + 2
\]

#### 2.2.3 \(g(x)\):

The values of \(g(x)\) are:
\[
\text{For } x = 1, g(x) = 5
\]
\[
\text{For } x = 2, g(x) = 11
\]
\[
\text{For } x = 3, g(x) = 19
\]
\[
\text{For } x = 4, g(x) = 29
\]
\[
\text{For } x = 5, g(x) = 41
\]

To find the relationship, observe the differences:
\[
\text{From } g(1) \text{ to } g(2): 11 - 5 = 6
\]
\[
\text{From } g(2) \text{ to } g(3): 19 - 11 = 8
\]
\[
\text{From } g(3) \text{ to } g(4): 29 - 19 = 10
\]
\[
\text{From } g(4) \text{ to } g(5): 41 - 29 = 12
\]

The differences are increasing by 2 each time, indicating a quadratic relationship.

Given the pattern, the function could be modeled as:
\[
g(x) = x^2 + 3x + 1
\]

This can be verified by substituting the values of \(x\) into the quadratic function:
- \(g(1) = 1^2 + 3(1) + 1 = 1 + 3 + 1 = 5\)
- \(g(2) = 2^2 + 3(2) + 1 = 4 + 6 + 1 = 11\)
- \(g(3) = 3^2 + 3(3) + 1 = 9 + 9 + 1 = 19\)
- \(g(4) = 4^2 + 3(4) + 1 = 16 + 12 + 1 = 29\)
- \(g(5) = 5^2 + 3(5) + 1 = 25 + 15 + 1 = 41\)

Thus, the correct function for \(g(x)\) is:
\[
g(x) = x^2 + 3x + 1
\]

---

Would you like further explanations on any of these points or help with another topic?

### Related Questions:
1. How do you convert a general equation into function form?
2. What is the significance of the slope and intercept in a linear function?
3. How can you determine if a set of points follows a linear or quadratic function?
4. What methods can you use to find the equation of a function from a table of values?
5. How would \(f(x)\) change if the rate of increase was doubled?
6. Can a quadratic function have a constant difference between terms? Why or why not?
7. What is the role of coefficients in determining the shape of a quadratic function?
8. How do you solve for the roots of a quadratic function like \(g(x)\)?

### Tip:
When analyzing functions, always check for patterns in the differences between values. Constant differences suggest a linear function, while changing differences often indicate a quadratic or higher-degree polynomial function.

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