For continuity at x=-1, ax + b must equal 2 at x=-1
Therefore -a + b = 2
For continuity at x=3, ax+b must equal -2 there.
Therefore 3a + b = -2
Solve those two equations in two unknowns for a and b.
4a = -4
a = -1
b = 2 + a = ?
f(x) = 2 , x=<-1
ax + b , -1<x<3
-2 , x>=3
could someone help me with this question please? thank you.
Therefore -a + b = 2
For continuity at x=3, ax+b must equal -2 there.
Therefore 3a + b = -2
Solve those two equations in two unknowns for a and b.
4a = -4
a = -1
b = 2 + a = ?
To check the continuity at x = -1, we need to ensure that the left-hand limit as x approaches -1 is equal to the right-hand limit as x approaches -1.
Taking the left-hand limit:
lim(x->-1-) (ax + b) = -a + b
Taking the right-hand limit:
lim(x->-1+) (ax + b) = -a + b
The left-hand limit should be equal to the right-hand limit, so we have: -a + b = 2.
To check the continuity at x = 3, we need to ensure that the left-hand limit as x approaches 3 is equal to the right-hand limit as x approaches 3.
Taking the left-hand limit:
lim(x->3-) (ax + b) = 3a + b
Taking the right-hand limit:
lim(x->3+) (-2) = -2
The left-hand limit should be equal to the right-hand limit, so we have: 3a + b = -2.
Now we have the following system of equations:
-a + b = 2 (equation 1)
3a + b = -2 (equation 2)
Solving this system of equations, we can add equation 1 and equation 2 together:
(-a + b) + (3a + b) = (2) + (-2)
2a + 2b = 0
Divide both sides by 2:
a + b = 0
Now, substitute this expression for b into equation 1:
-a + (0) = 2
-a = 2
Divide by -1:
a = -2
Substituting this value of a back into the expression for b in equation 1:
-2 + b = 2
b = 4
So, the constants a = -2 and b = 4 make the function f continuous on the entire real line.