Understanding Piecewise Functions and Continuity

The video tutorial explains how to evaluate a piecewise 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 3. The tutorial analytically examines whether f(3) is defined, if the limit as x approaches 3 exists, and if the function is continuous at x=3. It concludes that f(3) is defined, but the limit does not exist, and the function is not continuous at x=3. These results are verified graphically, showing a break in the graph at x=3.

What is the value of f(x) when x is exactly 3 in the given piecewise function?

• Undefined

• -4

• Square root of (x - 2)

• 2x - 3

Which function rule applies when x is less than 3?

• 2x - 3

• Square root of (x - 2)

• -4

• x^2

What must be true for the limit of f(x) as x approaches 3 to exist?

• The function must be defined at x = 3

• The left-hand and right-hand limits must be equal

• The function must be differentiable at x = 3

• The function must be continuous at x = 3

What is the left-hand limit of f(x) as x approaches 3?

• 1

• 0

• 3

• -4

What is the right-hand limit of f(x) as x approaches 3?

• 3

• 0

• 1

• -4

Why does the limit of f(x) as x approaches 3 not exist?

• The function is not continuous at x = 3

• The function is not defined at x = 3

• The function is not differentiable at x = 3

• The left-hand and right-hand limits are not equal

What condition must be met for a function to be continuous at a point?

• The function must be defined at that point

• The limit must exist at that point

• The limit must equal the function value at that point

• All of the above

Is the function continuous at x = 3?

• Yes, because the function is defined at x = 3

• No, because the function is not differentiable at x = 3

• No, because the limit does not exist at x = 3

• Yes, because the left-hand and right-hand limits are equal

What does the graph of the function show at x = 3?

• A continuous line

• A hole at x = 3

• A jump discontinuity

• A vertical asymptote

What is the function value at x = 3 on the graph?

• 1

• 3

• Undefined

• -4