To find the constant 'a' for the given piecewise function to ensure it's continuous, we must ensure the two parts of the function are equal at x = 1.
This function is divided into two parts:
- f(x) = 3x^2 for values x >= 1
- f(x) = ax + 5 for values x < 1
The point of intersection for these two functions is at the limiting boundary, which is x = 1. Hence, for the function to be continuous, the output of both these parts must be equal at x = 1.
Let's calculate the output for each function at x = 1:
- For f(x) = 3x^2, the output at x = 1 is then f(1) = 3*1^2 = 3
- For f(x) = ax + 5, the output at x = 1 is simply a*1 + 5 = a + 5
For the function to be continuous, these two outputs must be equal. So, we form the equation 3 = a + 5.
If we solve this equation for 'a' by subtracting 5 from both sides, we obtain a = 3 - 5, which simplifies to a = -2.
So, the constant 'a' that makes this function continuous everywhere is -2.