Understanding Function Continuity and Simplification

The video tutorial explains a function f(x) = (6x^2 + 18x + 12) / (x^2 - 4) and its undefined points at x = ±2 due to division by zero. It demonstrates simplifying the function by factoring and discusses constraints for continuity. The tutorial concludes by redefining the function to make it continuous at x = -2, assigning a value of 3/2.

Why is the function f(x) not defined at x = ±2?

• Because x = ±2 makes the numerator zero.

• Because x = ±2 makes the function negative.

• Because x = ±2 makes the denominator zero.

• Because x = ±2 makes the function positive.

What is the first step in simplifying the function f(x)?

• Subtracting a constant from the denominator.

• Adding a constant to the numerator.

• Factoring out a common factor from the numerator.

• Multiplying the denominator by a constant.

What is the result of factoring the denominator of f(x)?

• x - 2 times x + 1

• x + 2 times x - 2

• x - 2 times x - 2

• x + 2 times x + 2

Why must we assume x is not equal to -2 when simplifying f(x)?

• Because it makes the function undefined.

• Because it makes the denominator zero.

• Because it makes the function continuous.

• Because it makes the numerator zero.

What is the simplified form of f(x) after canceling common factors?

• 6 times x minus 1 over x minus 2

• 6 times x plus 1 over x minus 2

• 6 times x plus 1 over x plus 2

• 6 times x minus 1 over x plus 2

What value should be assigned to f(-2) to make f(x) continuous?

• 1/2

• 3/2

• 2/3

• 1/3

How do we redefine f(x) to ensure it is continuous at x = -2?

• Set f(x) to -1 for x = -2.

• Set f(x) to 1 for x = -2.

• Set f(x) to 3/2 for x = -2.

• Set f(x) to 0 for x = -2.

What is the equivalent expression of the redefined f(x)?

• 6 times x minus 1 over x minus 2

• 6 times x plus 1 over x minus 2

• 6 times x minus 1 over x plus 2

• 6 times x plus 1 over x plus 2

What is the purpose of redefining f(x) at x = -2?

• To make the function continuous.

• To make the function undefined.

• To make the function zero.

• To make the function negative.

What is the final conclusion about the redefined function f(x)?

• It is undefined at all points.

• It is not equivalent to the simplified expression.

• It is equivalent to the simplified expression.

• It is only defined at x = 2.