Final answer:
The problem is to find values of 'a' and 'b' that make a piecewise function continuous, which involves ensuring that the function has no breaks, jumps, or holes at the points where the different pieces of the function meet.
Step-by-step explanation:
The question is asking for values of a and b such that the piecewise function f(x) is continuous at every x. To be continuous, the function must not have any breaks, jumps, or holes in its graph. Particularly, we are looking at the continuity of three expressions:
- f(x) = (x² - 4) / (x - 2) for x not equal to 2.
- ax² - bx + 3 for some value of x.
- 32x - a + b for some value of x.
For the first expression to be continuous at x = 2, the limit as x approaches 2 must exist. That limit is equal to the value 4, which is obtained by factoring and cancelling out the common term (x - 2). The second and third expressions are polynomials, which are always continuous for all real numbers. The continuity of the piecewise function f(x) will depend on whether the adjoining parts of the function agree at their points of transition. So, the task is to find values of a and b that will ensure f(x) to be continuous at these transition points.