Question

Let $f$ be a linear function for which $f(6)-f(2)=12$. What is $f(12)-f(2)?$ Please explain how you found your answer. Thank you!

Answer 1

Answer: 30

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Step-by-step explanation:

Since f(x) is linear, this means f(x) = mx+b

• m = slope
• b = y intercept

Let's plug in x = 6

`f(x) = mx+b\\f(6) = m*6+b\\f(6) = 6m+b`

Repeat for x = 2

`f(x) = mx+b\\f(2) = m*2+b\\f(2) = 2m+b`

Now subtract the two function outputs

`f(6)-f(2) = (6m+b)-(2m+b)\\f(6)-f(2) = 6m+b-2m-b\\f(6)-f(2) = 4m\\`

The b terms cancel out which is very handy.

Set this equal to 12, since f(6)-f(2) = 12, and solve for m

`f(6)-f(2) = 12\\4m = 12\\m = 12/4\\m = 3\\`

So the slope of f(x) is m = 3

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Next, plug in x = 12

`f(x) = mx+b\\f(12) = m*12+b\\f(12) = 12m+b`

We can then say:

`f(12)-f(2) = (12m+b)-(2m+b)\\f(12)-f(2) = 12m+b-2m-b\\f(12)-f(2) = 10m\\`

Lastly, we plug in m = 3

`f(12)-f(2) = 10m\\f(12)-f(2) = 10*3\\f(12)-f(2) = 30\\`