Determine the constants a and b so that the function f is continuous on the entire real line:
f(x) = 2 , x=<-1
ax + b , -1<x<3
-2 , x>=3
could someone help me with this question please? thank you.
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 = ?
THANK YOU!!
To determine the constants a and b so that the function f is continuous on the entire real line, we need to ensure that the function is continuous at the points where the pieces of the function meet, which in this case is at x = -1 and x = 3.
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.