Understanding Piecewise Functions and Limits

The video tutorial explains how to analyze a piecewise defined function, f(x), by determining the truth of various statements about it. The function is defined differently for x less than, greater than, and equal to -4. The tutorial covers calculating limits from both sides of x = -4, checking the function's continuity, and verifying results graphically. It concludes with a review of the statements and a graphical representation of the function.

What is the value of f(x) when x is less than -4?

• 3x + 21

• x^2 + 1

• 2

• x + 85

What is the value of f(-4) according to the piecewise function?

• 0

• 9

• 2

• 3

What must be true for the limit as x approaches -4 to exist?

• The left-hand limit must equal the right-hand limit.

• The function must be continuous.

• The function value must be zero.

• The derivative must exist.

What is the limit of f(x) as x approaches -4 from the left?

• Undefined

• 2

• 0

• 9

What is the limit of f(x) as x approaches -4 from the right?

• 2

• 9

• Undefined

• 0

Is the function continuous at x = -4?

• No, because the function is not defined.

• Yes, because the limit exists.

• No, because the limit does not equal the function value.

• Yes, because the function is defined.

What graphical feature indicates the function is not continuous at x = -4?

• A vertical asymptote

• A sharp corner

• A horizontal line

• A hole in the graph

What is the function value at x = -4 on the graph?

• 9

• 0

• Undefined

• 2

What is the y-coordinate of the closed point at x = -4?

• Undefined

• 0

• 9

• 2

What is the significance of the hole in the graph at x = -4?

• It indicates a point of discontinuity.

• It shows the function is undefined.

• It represents a maximum point.

• It is a point of inflection.