Question

Have never seen a piecewise function quite like this before and am not sure how to even graph the mx+b line in all honesty.

Answer 1

Answer: d = 21/127, c = 45/127

Proof:

First simplify the first equation. At first glance it looks quadratic, but it is actually linear. f(x) = c + d((x²-64)/(x-8)) = c + dx + 8d

Next, you must find values for c and d that makes the following true:

1. f(8) exists

2. the limit as x approaches 8 from the left and the right it the same

3. that limit from above is the value of f(8)

To do this, first solve for a ratio between c and d by setting x to equal 8. Then, set the 1st and 3rd formulas equal to each other:

c + d(8) + 8d = c(8) + d

7c = 15d

c = (15/7)d

Now we can solve to see what c and d actually are. Make x = 8 and set the 2nd and 3rd functions equal to each other, but this time plug in the value of c we found into the 3rd equation. Using the 2nd function is easier than it looks, as when x = 8, the second equation is just 3.

3 = (15/7)d + 16d = (127/7)d

d = 21/127

Now use the formula 7c = 15d to solve for c

7c = 15(21/127)

c = (45/127)

We know these values of c and d are true because they satisfy all three requirements listed above.