Final answer:
To find the constant a for function g(x) to be continuous, factor and cancel terms, then set the limit as x approaches a equal to the function's value at x = a. The solution is a = 2. There is no requirement for a constant b mentioned in the problem.
Step-by-step explanation:
The student's question requires finding values for the constants a and b to ensure the function g(x) is continuous on the entire real line. A function is continuous if it can be drawn without lifting the pencil from the paper, which mathematically means the following two conditions must be met:
- The function is defined at the point.
- The limit of the function as it approaches the point from either direction must equal the function's value at that point.
To find the constant a, we simplify the given function g(x) when x ≠ a by factoring the numerator x2 - a2 as (x + a)(x - a) and canceling out the common (x - a) terms:
g(x) = (x2 - a2) / (x - a) = (x + a)
For g(x) to be continuous at x = a, the limit as x approaches a must equal 4 (since g(a) = 4):
limx→a g(x) = limx→a (x + a) = 2a
To make the function continuous, we set 2a = 4, solving for a gives us a = 2. There is no constant b mentioned in the original problem, so we assume it is a typo and b is not required for the solution.