Function Continuity Worksheets Grade 12

1. Emma is designing a roller coaster track that follows the piecewise function f(x) = { x^2 + 1 for x < 2, ax + b for 2 ≤ x ≤ 4, 3x - 1 for x > 4 }. To ensure a smooth ride, the track must be continuous at both transition points x = 2 and x = 4. What values must the parameters a and b have to guarantee continuity throughout the track?
2. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (5t^2 * e^(-0.3t))/(t^2 + 1), where t is measured in hours. The researchers need to determine if this concentration function is continuous for all t ≥ 0, particularly at t = 0 where the function appears to have an indeterminate form. Analyze the continuity of C(t) at t = 0 and explain your reasoning mathematically.
3. Consider the function f(x) = (x^2 - 4)/(x - 2) for x ≠ 2. Determine the value that f(2) should be assigned to make the function continuous at x = 2.

…and 3 more problems

1. Given the graph of f(x) with points at (9,4), (10,4), (11,6), and a jump discontinuity at x=10 where f(10) = 4 but lim(x→10⁻) f(x) = 4 and lim(x→10⁺) f(x) = 6. Is f(x) continuous at x=10?
2. Given the graph of function f(x) with a jump discontinuity at x = 3, a removable discontinuity at x = 5, and an infinite discontinuity at x = 7. Which x-values represent points of discontinuity and what type is each?
3. A continuous function f(x) is graphed on the coordinate plane. The graph passes through points (-2, 4), (0, 1), and (3, -2). Between x = -2 and x = 3, the function is a smooth curve with no breaks, jumps, or holes. According to the Intermediate Value Theorem, what is the minimum number of times the function must cross the x-axis between these endpoints?

…and 4 more problems

1. A function f(x) is graphed on the coordinate plane. The graph consists of a smooth curve from (-3, 2) to (1, 4), but has a hole at x = 1 where the curve would pass through (1, 4) if extended. From x = 1 to x = 4, the graph is a different smooth curve starting at (1, 2) and ending at (4, 6). At x = 2, there is a vertical asymptote where the function approaches positive infinity from both sides. Identify all x-values where the function is discontinuous and classify each discontinuity.
2. Sophia is analyzing the continuity of f(x) from its graph. The graph shows: f(1) = 6, lim(x→1⁻) f(x) = 6, lim(x→1⁺) f(x) = 6. Is f(x) continuous at x = 1?
3. Given the graph of Sophia's function f(x) with points at (-1,1), (1,2), (3,1), (5,0), (6,1), identify all x-values where the function is discontinuous and classify each discontinuity type.

…and 5 more problems