To determine the correct function, we can use the conditions given:

1. $f(8) = 16$
2. $f(-3) = -17$

Assuming the function is in the form $f(x) = ax + b$, we can set up two equations based on these conditions:

For $f(8) = 16$: $8a + b = 16$

For $f(-3) = -17$: $-3a + b = -17$

Now, let’s solve these two equations simultaneously.

1. From the first equation: $8a + b = 16$
2. From the second equation: $-3a + b = -17$

Subtracting the second equation from the first to eliminate $b$: $(8a + b) - (-3a + b) = 16 - (-17)$ $11a = 33$ $a = 3$

Now, substitute $a = 3$ into one of the original equations to find $b$: $8(3) + b = 16$ $24 + b = 16$ $b = -8$

Thus, the function is: $f(x) = 3x - 8$

Looking at the options, the correct answer is: $\boxed{f(x) = 3x - 8}$

Would you like further details on this solution?

Here are 5 related questions to expand understanding:

1. How would the solution change if the function were quadratic instead of linear?
2. Can we generalize this approach to find any linear function given two points?
3. What other values of $f(x)$ could confirm the function's linearity?
4. How would the answer change if one of the conditions was incorrect?
5. What if $f(x)$ were given as a different form, like $f(x) = mx + c$?

Tip: When dealing with function values at specific points, setting up simultaneous equations based on the function's form helps isolate coefficients.