Understanding Discontinuities in Graphs

The video tutorial covers the concept of continuity in graphs, explaining different types of discontinuities such as holes, jump discontinuities, and infinite discontinuities. It provides methods to identify these discontinuities in rational functions and piecewise functions. The tutorial includes examples to illustrate these concepts and demonstrates how to solve for constants to ensure continuity in piecewise functions.

Which type of discontinuity is characterized by a gap where the graph does not connect?

• Hole

• Jump

• Infinite

• Removable

How can you find the vertical asymptote in a rational function?

• By finding the x-intercept

• By setting the numerator equal to zero

• By finding the y-intercept

• By setting the denominator equal to zero

What type of discontinuity is associated with a vertical asymptote?

• Hole

• Removable

• Jump

• Infinite

In the function f(x) = |x|/x, what type of discontinuity is present?

• Hole

• Jump

• Infinite

• Removable

What is the only type of removable discontinuity?

• None

• Infinite

• Jump

• Hole

For a piecewise function, where are discontinuities most likely to occur?

• At the minimum points

• At the maximum points

• At the points where the function pieces meet

• At the endpoints of the domain

How do you determine if a piecewise function is continuous at a point?

• Check if the function is integrable at that point

• Check if the y-values of the function pieces are equal at that point

• Check if the function has a vertical asymptote at that point

• Check if the function is differentiable at that point

What value of C makes the function f(x) = Cx + 3 (x < 2) and 3x + C (x ≥ 2) continuous at x = 2?

• 1

• 2

• 4

• 3

If f(x) = ax - 2 (x < 3) and x^2 - 5 (x ≥ 3), what value of a makes the function continuous at x = 3?

• 4

• 3

• 2

• 1

For the function f(x) = ax + 5 (x < 1), x^2 - Bx + 9 (1 ≤ x < 4), and ax^2 - Bx - 7 (x ≥ 4), what are the values of A and B for continuity?

• A = 4, B = 5

• A = 3, B = 4

• A = 2, B = 3

• A = 1, B = 2