Find the constant a such that the function is continuous on the entire real line.
f(x) =
5x^2, x ≥ 1
ax − 3, x < 1
???Please, thnks.
To find the constant "a" such that the function is continuous on the entire real line, we need to make sure that the two pieces of the function, 5x^2 for x ≥ 1 and ax - 3 for x < 1, connect smoothly at the point where x = 1.
For a function to be continuous at a point, the limit of the function as x approaches that point from both sides should be equal, and the function value at that point should also be equal to this limit.
Let's calculate the limit of the function as x approaches 1 from the left (x < 1):
lim(x→1-) ax - 3 = a(1) - 3 = a - 3
Now, let's calculate the limit of the function as x approaches 1 from the right (x ≥ 1):
lim(x→1+) 5x^2 = 5(1^2) = 5
Since the function should be continuous at x = 1, the two limits should be equal:
a - 3 = 5
Now, solving this equation for "a":
a = 5 + 3 = 8
Hence, the constant "a" that makes the function f(x) continuous on the entire real line is a = 8.