Function Continuity Worksheets Grade 12

1. Charlotte, a civil engineer, is examining the graph of a function f(x) that models the stress distribution (in megapascals) along a steel girder, where x is the distance in meters from the left support. The graph shows that as x approaches 9 from the left, f(x) approaches 16, and as x approaches 9 from the right, f(x) approaches 16. However, at x = 9, the graph has an open circle at y = 16 and a separate closed dot at (9, 22). Based on this graphical information, is the function continuous at x = 9? If not, identify the type of discontinuity and explain your reasoning using the conditions for continuity.
2. Given the graph of Matiu's function f(x) with points at x=4, x=8, and x=12, determine if f(x) is continuous at each point and classify any discontinuities: f(4)=12, lim_(x→4⁻) f(x)=12, lim_(x→4⁺) f(x)=12; f(8)=20, lim_(x→8⁻) f(x)=16, lim_(x→8⁺) f(x)=16; f(12)=24, lim_(x→12⁻) f(x)=24, lim_(x→12⁺) f(x)=28.
3. Given the graph of Emma's function f(x) with points at x=1, x=3, x=5, x=7, and x=9, determine if f(x) is continuous at each point and classify any discontinuities: f(1)=3, lim_(x→1⁻) f(x)=3, lim_(x→1⁺) f(x)=3; f(3)=7, lim_(x→3⁻) f(x)=5, lim_(x→3⁺) f(x)=5; f(5)=9, lim_(x→5⁻) f(x)=9, lim_(x→5⁺) f(x)=11; f(7)=13, lim_(x→7⁻) f(x)=13, lim_(x→7⁺) f(x)=13; f(9)=15, lim_(x→9⁻) f(x)=15, lim_(x→9⁺) f(x)=17.

…and 3 more problems

1. 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?
2. A pharmaceutical company is modeling the concentration of a new drug in the bloodstream over time. The concentration function is given by C(t) = (t² - 4)/(t - 2) for t ≠ 2, where t is measured in hours. The researchers need to determine if the concentration function is continuous at t = 2 hours to ensure predictable drug effects. If not continuous, they must identify the type of discontinuity present.
3. 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?

…and 4 more problems

1. Mason is an electrical engineer analyzing a signal in a circuit. The signal's voltage (in volts) as a function of time t (in seconds) is modeled by the piecewise function V(t) = { t^2 - 16 for t < 4, a*t + b for 4 ≤ t ≤ 7, 3*t + 1 for t > 7 }. For the signal to be transmitted without distortion, the voltage function must be continuous at both transition points t = 4 seconds and t = 7 seconds. What values must the parameters a and b have to ensure the voltage function is continuous throughout the circuit?
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. Maria is designing a roller coaster track that follows the piecewise function f(x) = { x^2 + 2 for x < 1, ax + b for 1 ≤ x ≤ 3, 4x - 2 for x > 3 }. To ensure a smooth ride, the track must be continuous at both transition points x = 1 and x = 3. What values of the parameters a and b will make the roller coaster track continuous at both points?

…and 3 more problems