For the linear function f(x) = ax + b. if f(-3) = -1 and f(x + 1) - f(x - 1) = 28, What is the value of f(2)!

Question

This is a linear function that was given to me by my teacher and I don't no how to answer it so can u please help me 

Answer 1

Katelyn,

 

Your strategy for solution should be: first determine the constants (a & b) of the linear function using the data points given in the problem; then once you have the function fully specified, evaluate it at x = 2.

 

f(x) = ax + b

From the condition f(-3) = -1 you get the equation:  a(-3) + b = -1 , or  -3a + b = -1.

From the other condition {f(x+1)} -  {f(x-1)} = 28, you get a second equation in a & b

{a(x+1) + b} - {a(x-1) + b} = 28

ax + a + b -ax + a - b,  or 2a = 28  so   a = 14

Then from the first relationship, -3(14) + b = -1  you get b = -1 + 42 = 41

 

So now the original equation becomes:

f(x) = ax + b  = 14x + 41

 

And f(2) = 14(2) + 41 = 28 + 41 = 69.

 

You can recheck the data they gave you to make sure your equation is correct:

f(-3) = 14(-3) + 41 = -42 + 41 = -1 so it checks here

f(x+1) = 14(x+1) + 41 = 14x + 55

f(x-1)  = 14(x-1) + 41  =  14x + 27

Then {f(x+1) - f(x-1)} = 14x - 14x + 55 - 27  = 28 and this checks out also,

so now you can be sure that f(2) = 69.

 

BTW,  you should recognize an old friend, y = mx + b as the slope-intercept way of writing that linear equation, where m = a is the slope of the line and b the y-intercept when x=0.

 

The second condition let's you quickly compute the slope m = Δf(x)/Δx = {f(x+1) - f(x-1)}/2 = 28/2 = 14 = m = a.

And compute b from the first condition,  b = y - 14x = -1 -14(-3) = 41.

Then compute y = 14(2) + 41 = 69.

Answer 2

Katelyn,

 

You are give that f(x) = ax + b and that f(-3) = -1

Substitute -3 for x in the first formula so:

 

f(-3) = a(-3) + b = -3a + b = -1

 

We now know that:  -3a + b = -1

 

Now we need to work with f(x + 1) - f(x - 1) = 28

 

Substituting (x + 1) for x in the first formula gives:

 

f(x + 1) = a(x + 1) + b

 

and with (x-1) we get:

 

f(x - 1) = a(x -1) + b

 

Subtracting the last from the former gives:

 

(ax + a + b) - (ax - a +b) = 28

 

simplifying:  ax + a + b - ax +a -b = 2a = 28.

 

so a = 28/2 = 14  

 

We can use this in our expression: -3a + b = -1 to get:

 

-3(14) + b = -1 ⇒ -42 + b = -1 ⇒ b = -1 +42 = 41

 

so our function is:  f(x) = 14x + 41

 

then f(2) = 14(2) + 41 = 28 + 41, so 

 

f(2) = 69

 

 

 

 

 

 

Answer 3

Since f(-3) = -1, -3a + b = -1, so b = 3a - 1

 

f(x + 1) - f(x - 1) = 28

 

a(x + 1) + b - [a(x - 1) +b] = 28

 

ax + a + b - ax + a - b = 28

 

So, 2a = 28.  Therefore, a = 14 and b = 3a - 1 = 3(14) - 1 = 41

 

We have f(x) = 14x + 41.   So, f(2) = 14(2) + 41 = 28 + 41 = 69