how do I write a rule for this function?

x: -4, 0, 2, 4
f(x): 4, 6, 7, 8

slope = (8-4)/(4--4) = 4/8 = 1/2

It is constant for all points
so
y = .5 x + b
8 = .5(4) + b
6 = b
so
y = .5 x + 6

check that for all points like (-4,4)
4 = (.5)(-4) + 6 ???
4 = -2 + 6
4 = 4 sure enough

To write a rule for a function, you need to find the relationship between the input (x) and the output (f(x)). In this case, analyze the given values for x and f(x) to determine how they are related.

Looking at the given values, notice that as the value of x increases by 2, the value of f(x) increases by 2 as well. This suggests a linear relationship between x and f(x), with a constant rate of change.

To find the rate of change, calculate the difference in f(x) for each pair of consecutive x-values:

f(0) - f(-4) = 6 - 4 = 2
f(2) - f(0) = 7 - 6 = 1
f(4) - f(2) = 8 - 7 = 1

It is clear that the difference in f(x) is always 1, regardless of the value of x. Therefore, the rule for this function can be expressed as f(x) = x + 5.

To verify this rule, try plugging in the given x-values:

For x = -4: f(-4) = -4 + 5 = 1
For x = 0: f(0) = 0 + 5 = 5
For x = 2: f(2) = 2 + 5 = 7
For x = 4: f(4) = 4 + 5 = 9

As you can see, the values obtained from the rule match the given values for f(x), confirming that the rule f(x) = x + 5 correctly represents the relationship between x and f(x) in this function.