AlgebraVery hardLinear functions from a constant difference condition
Algebra practice question
The function f is linear, and for every value of x, f(x + 2) − f(x) = 6. If f(5) = 4, what is the value of f(0)?
- A. −26
- B. −11Correct
- C. 19
- D. 34
Answer: B. −11
For a linear function f(x) = mx + c, f(x + 2) − f(x) = 2m. So 2m = 6 and m = 3. Moving from x = 5 to x = 0 decreases x by 5, so f(0) = f(5) − 5(3) = 4 − 15 = −11. (Equivalently, 4 = 3(5) + c gives c = −11, and f(0) = c.) Choice A treats 6 as the slope itself instead of the change over 2 units: 4 − 5(6) = −26. Choice C uses the correct slope but the wrong direction, adding instead of subtracting: 4 + 15 = 19. Choice D combines both errors: 4 + 5(6) = 34.
Why the other answers are wrong
- A. −26
- You used 6 as the slope, but 6 is the change over 2 units of x, so the slope is 3; 4 − 5(6) = −26 instead of 4 − 5(3) = −11.
- C. 19
- You used the right slope of 3 but went the wrong direction: moving from x = 5 to x = 0 decreases x, so subtract 15 rather than adding it.
- D. 34
- This combines both slips — treating 6 as the slope and adding instead of subtracting: 4 + 5(6) = 34, when it should be 4 − 5(3).
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