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I have a solution here to the same problem but a slight different given:

-1 x<=-1 
ax-b -1<x<1 
2 x>=1 

The solution is as follows:

 f(x) = {-1, x ≤ -1 
...... { ax - b, -1 < x < 1 
...... { 2, x ≥ 1 


Then we must have: 
lim(x-->-1) -1 = lim(x-->-1) ax - b 
==> -1 = -a - b 
==> a + b = 1 

And we must have: 
lim(x-->1) ax - b = lim(x-->1) 2 
==> a - b = 2 

So we need to solve the system of equations given by: 
a + b = 1 
a - b = 2 

After doing so, you should get that: 
a = 3/2 
b = -1/2 

By studying the solution above, you now have the ability to answer the problem on your own! 



facundo3141592

By solving a system of equations, we will see that the function is continuous for all values of x if:

  • a = 2.5
  • b = -0.5.

How to make the piecewise-function continuous for every x?

Here we have the piecewise function:

f(x) = -7            if x ≤  -3

f(x) = ax- b     if -3 < x < 1

f(x) = 3             if  1 ≤ x

The piecewise function will be continuous only if the values at the limits where we have jumps (at x = -3 and x = 1) are the same in both pieces of the function.

This means that we must have:

f(-3) = -7 = a(-3) - b

f(1) =  3 = a*1 - b

Then we have a system of equations:

-7 = -3a - b

3 = a - b

To solve this, first, we isolate one of the variables in one of the equations, I will isolate b in the second one to get:

b = a - 3

Now we can replace that in the other equation to get:

-7 = -3a - (a - 3)

Now we can solve this for a.

-7 = -4a + 3

-7 -3 = -4a

-10/-4 = a = 2.5

Then the value of a must be 2.5, to get the value of b we use:

b = a - 3 = 2.5 - 3 = -0.5

So the solution is a = 2.5, b = -0.5.

If you want to learn more about systems of equations, you can read:

https://brainly.com/question/13729904

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