Final answer:
The question involves determining the continuity of a piecewise function f(x). The function has different expressions on either side of x = 1 and is not continuous at x = 1 because the left-hand limit and the right-hand limit of f(x) as x approaches 1 are not equal.
Step-by-step explanation:
The question asks us to consider two pieces of the function f(x) and to determine whether f(x) is continuous. The function is defined differently on either side of x = 1:
- For x ≤ 1, f(x) = -x + 2
- For x > 1, f(x) = 5x - 2
To determine continuity at x = 1, both the left-hand limit and right-hand limit as x approaches 1 must be equal to the function value at x = 1.
Computing the left-hand limit as x approaches 1, f(x) = -1 + 2 = 1. Computing the right-hand limit as x approaches 1, f(x) = 5(1) - 2 = 3. Since the limits are not equal, the function is not continuous at x = 1, indicating that f(x) is not continuous over the entire domain.
To graph f(x) over its entire domain, plot the piecewise linear components on a coordinate plane, showing a break in the function at x = 1.