Final answer:
The constant 'a' required for the function g(x) to be continuous on the entire real line is 0, as it makes the limit and the function's value at x = a² consistent.
Step-by-step explanation:
To find the constant a that makes the function g(x) = x² - a²x - a continuous on the entire real line, we need to ensure that the function is defined at x = a² and that there is no jump discontinuity at that point. Since continuous functions have limits that equal the function's value at the point, we must find a such that ℝ
limx → a²g(x) = g(a²).
Because the function g(x) is not defined at x = a² in the original statement, we will replace x with a² and therefore:
g(a²) = (a²)² - a²(a²) - a.
This simplifies to:
g(a²) = a⁴ - a⁴ - a = -a.
Now that we have the value of the function at x = a², we set the limit equal to this value:
limx → a²g(x) = -a.
As x approaches a², the function becomes:
g(x) = x² - a²x - a, which for the limit as x approaches a² simplifies to:
x² - a²x - a.
Now we equate the limit to the function's value at x = a²:
(a²)² - a²(a²) = -a
a⁴ - a⁴ = -a
This results in 0 = -a. Therefore, the constant a is 0.