define f(x)={x+a if x>2

{-1 if x=2
{ax+b if x<2.
How to determine the values of a and b for which f(x) is continuous in the set of real numbers.

see your other post for the method.

That is,

http://www.jiskha.com/display.cgi?id=1485688337

To determine the values of "a" and "b" for which the function f(x) is continuous in the set of real numbers, we have to ensure that the function is continuous at the point where the conditions change, which is x = 2 in this case.

In order for f(x) to be continuous at x = 2, the left-hand limit and the right-hand limit must be equal to the value of the function at x = 2.

1. Left-hand limit (x → 2⁻): We evaluate f(x) as x approaches 2 from the left side.
Since f(x) = ax + b for x < 2, we substitute x = 2 into the function and calculate the limit:
lim(x → 2⁻) f(x) = lim(x → 2⁻) (ax + b) = 2a + b

2. Right-hand limit (x → 2⁺): We evaluate f(x) as x approaches 2 from the right side.
Since f(x) = x + a for x > 2, we substitute x = 2 into the function and calculate the limit:
lim(x → 2⁺) f(x) = lim(x → 2⁺) (x + a) = 2 + a

3. Continuity at x = 2: Since f(x) = -1 when x = 2, the value of the function at x = 2 should be -1.
So, f(2) = -1.

Now, we set the left-hand limit, right-hand limit, and the value of the function equal to each other and solve for "a" and "b":
2a + b = 2 + a = -1

To solve this system of equations, subtract a from both sides of the second equation:
2a + b = 2 + a
a + b = -1 - a
2a + b = -1

Subtract a from both sides of the third equation:
-1 - a = -1
a = 0

Substitute a = 0 into the second equation:
0 + b = -1 - 0
b = -1

Therefore, for the function f(x) to be continuous in the set of real numbers, the values of "a" and "b" should be a = 0 and b = -1.