Defining a function at a point to make it continuous | Limits | Differential Calculus | Khan Academy

TL;DR

The given function is not defined at x = 2, so the value of f(-2) needs to be assigned as 3/2 to make the function continuous.

Transcript

The function, f of x is equal to 6x squared plus 18x plus 12 over x squared minus 4, is not defined at x is equal to positive or negative 2. And we see why that is, if x is equal to positive or negative 2 then x squared is going to be equal to positive 4, and 4 minus 4 is 0, and then we're going to have a 0 in the denominator. And that's not define... Read More

Key Insights

• ☺️ The given function is not defined at x = 2 due to a denominator of 0.
• ☺️ By factoring and simplifying the function, it can be written as 6(x + 1)/(x - 2) with the constraint that x ≠ -2.
• ☺️ To make the function continuous at x = -2, the value of f(-2) needs to be assigned as 3/2.

Questions & Answers

Q: Why is the function not defined at x = 2?

The function is not defined at x = 2 because it leads to a denominator of 0, which is undefined mathematically.

Q: How can the function be simplified?

The function can be simplified by factoring out a 6 from the numerator and using the difference of squares in the denominator.

Q: Why is there a constraint on x ≠ -2 in the simplified function?

The constraint x ≠ -2 is necessary because if x = -2, the denominator would be 0, resulting in an undefined value for the function.

Q: How can the function be made continuous at x = -2?

To make the function continuous at x = -2, the value of f(-2) needs to be assigned as 3/2, which is obtained by evaluating the simplified function at x = -2.

Summary & Key Takeaways

• The function f(x) is not defined at x = 2 due to a denominator of 0.

• By simplifying the function, it can be rewritten as 6(x + 1)/(x - 2), with the constraint that x ≠ -2.

• To make the function continuous at x = -2, the value of f(-2) needs to be assigned as 3/2.