Let's break down and solve the problems in the image.

1. Identifying the types of functions and their equations:

In the first part of the problem (a) to (f), you are tasked with identifying the type of function (linear or exponential) and writing its corresponding equation.

(a) Function Type: Linear

You can see that the values in the table increase by 3 each time for $x$, so it fits a linear pattern.

• Equation: $y = 3x + 7$

(b) Function Type: Exponential

The values for $y$ follow an exponential pattern, doubling each time the $x$ value increases. The equation is written as:

• Equation: $y = -2(2^x)$

(c) Function Type: Exponential

The values of $y$ are divided by 2 each time the $x$ increases by 1, indicating an exponential decay:

• Equation: $y = 32(1/2)^x$

(d) Function Type: Linear

The values decrease by 16 for each unit increase in $x$, making this a linear relationship.

• Equation: $y = -16x$

(e) Function Type: Exponential

The values of $y$ grow exponentially by a factor of 4/3. The equation is in the form:

• Equation: $y = \frac{1}{16} \left(\frac{4}{3}\right)^x$

2. Modeling the given data set:

You are given the table:

The pattern for $y$ shows that it's being multiplied by 2 each time $x$ increases. This indicates exponential growth. The correct model is:

• Option (1): $y = 5(2)^x$

Let me know if you want more details or explanations on any part!

Here are 5 related questions to explore:

1. How do you distinguish between a linear and an exponential function using a table of values?
2. What is the general form of a linear equation?
3. What is the general form of an exponential equation?
4. Can exponential functions ever represent decay? How?
5. What methods can be used to find the equation from a table of values?

Tip: For exponential functions, pay attention to how the values change by constant multiplication, unlike linear functions where changes are by constant addition.