An engraver charges a $10 fee plus $6 for each line of engraving. How do you write a function to describe this? What is a reasonable domain and range for the function for up to 8 lines of engraving?

1 Answer

#f(x) = 6x + 10#, where x represents the number of lines engraved.

Domain: #0 <= x <=8#
Range: #10 <= y <= 58#

Explanation:

In a y = mx + b equation, the $10 flat rate would represent b, the constant-- because the $10 never changes. Since the cost will change depending on how many lines are engraved, our m value will be 6.

y is another way of writing f(x). So that's how we get #f(x) = 6x + 10#.

For our domain, you mentioned that it must be reasonable to up to 8 lines. Since x represents the number of lines, we can say that our range must be less than or equal to 8. Of course, we cannot engrave a negative number of lines, which is why this would translate to #0 <= x <=8#.

For our range, the cost cannot sink below $10-- because even if you engrave 0 lines, the $10 flat rate will still remain. 8 engraved lines will produce a result of $58; #6(8) + 10 = 58#. Keeping this in mind, our range then must be #10 <= y <= 58#.

Hope this helps!

Best wishes,
A highschool student