In order to make the function continuous, we need to ensure that the limit of f(x) as x approaches a certain value from the left (x^-) is equal to the limit of f(x) as x approaches that value from the right (x^+).
1. Firstly, let's find the value of a. For this, we need to look at the point x=-1.
The function is defined in a piecewise way, such that for x<-1, f(x) = 3x⁻¹, and for -1≤x≤1/2, f(x) = ax.
In order to ensure continuity at x=-1, we need the limit of 3x⁻¹ as x approaches -1 from the left to be equal to the limit of ax as x approaches -1 from the right.
Let's set up an equation using the formulas for the limits:
lim_(x->-1^-) 3x⁻¹ = lim_(x->-1^+) ax
which simplifies to:
3(-1)⁻¹ = a(-1)
By solving this, we find that a = -3.
2. Secondly, we need to find b for continuity at x=1/2. Here the pieces of the function are f(x) = ax for -1≤x≤1/2, and f(x) = bx for x>1/2.
Again setting up an equation using the formulas for the limits:
lim_(x->1/2^-) ax = lim_(x->1/2^+) bx
which simplifies to:
a(1/2) = b(1/2)
Recall that we previously solved and found a = -3, substituting that in the equation provides us with:
-3(1/2) = b(1/2)
By solving this, we find that b = 2.
So the values of a and b that make the function continuous are a = -3 and b = 2.