Understanding Continuity in Piecewise Functions

The video tutorial explains how to determine the value of a constant, B, to make a piecewise function continuous everywhere. It highlights that the function is only potentially discontinuous at x = -1. The tutorial uses graphical analysis to illustrate continuity and demonstrates solving for B by equating the two function rules at x = -1. The solution shows that B must be -5 for the function to be continuous, and this is verified graphically.

What is the main goal of determining the value of B in the piecewise function?

• To identify the domain of the function

• To determine the slope of the function

• To find the maximum value of the function

• To make the function continuous everywhere

Why is the function potentially discontinuous at x = -1?

• Because the function is exponential

• Because the function is constant

• Because the function is piecewise defined

• Because the function is quadratic

What is the graphical indication of a discontinuity in a function?

• A horizontal line

• A smooth curve

• A break in the graph

• A sharp peak

What must be true for the two function rules at x = -1 to ensure continuity?

• They must be parallel

• They must have the same y-intercept

• They must be equal to each other

• They must have different slopes

What equation is used to find the value of B?

• 5x + 3 = -3x + B

• 5x - 3 = 3x + B

• 5x + B = -3x + 3

• 5x + 3 = 3x - B

What is the value of B that makes the function continuous?

• 5

• -3

• -5

• 3

What is the function rule for x > -1 after determining B?

• f(x) = 5x + 3

• f(x) = -3x - 5

• f(x) = 3x - 5

• f(x) = -5x + 3

How is the continuity of the function verified graphically?

• By checking for breaks in the graph

• By ensuring the graph is a straight line

• By finding the maximum point on the graph

• By checking the slope of the graph

What is the y-intercept of the function for x > -1?

• -3

• -5

• 5

• 3

What does the example demonstrate about piecewise functions?

• They are always continuous

• They have no real-world applications

• They can be made continuous by adjusting parameters

• They are never continuous