Find the values of a and b that make f continuous everywhere. f(x)={ (x^2-4)/(x-2) if x<

Find the values of a and b that make f continuous...

Key Concepts

Continuity in Piecewise Functions

This concept involves ensuring that a function defined by different expressions on different intervals does not have any jumps or breaks at the points where the definition changes. To be continuous, the limit from the left and the limit from the right at each transition point must be equal to the function's value at that point.

One-Sided Limits

One-sided limits are the values that a function approaches as the independent variable gets arbitrarily close to a specific point from either the left side or the right side. In the context of piecewise functions, it is essential to calculate the left-hand limit and the right-hand limit at the boundaries where the function's rule changes to ensure they coincide.

Removable Discontinuities

A removable discontinuity occurs when a function is not defined at a point, yet the limit as the function approaches that point exists. In many cases, such discontinuities can be 'fixed' or removed by appropriately redefining the function at that point, ensuring the overall function becomes continuous.

Setting Up and Solving Equations for Continuity Conditions

Ensuring the continuity of a piecewise function often involves equating the one-sided limits at the points of interest, which leads to the formation of equations involving any parameters in the function. Solving these equations simultaneously helps in finding the parameter values that guarantee continuity across the entire domain.

Transcript

00:01 Well let's now consider this function f of x equal x squared minus 4 over x minus 2 if x is smaller than 2 is equal to a x squared minus bx plus 3 if 2 is 2 is as a equal x or equal sorry just smaller than three and it's equal to two x minus a plus b if x is greater or equal three well as you can see there are two values two constants a and me let's try to find the values of these two constants a and b which make f continuous everywhere.

01:34 Everywhere.

01:35 So let's have a look at the function.

01:39 Well, the three different functions are continuous everywhere.

01:49 Well, just the first one is a rational function, and it's not continuous at x equal to, which doesn't belong to its domain.

02:03 So the points where we may have issues about continuity are the points as usual where the function breaks in some way, where the domain of the function breaks.

02:19 So they are x equal 2 and x equal 3.

02:29 Let's see what happens.

02:32 If we want this function continuous at x equal 2, what should we say? well, first of all, f of 2.

02:44 Well, f of 2 is equal to 4a.

02:52 We have to take into account the second function, so for x equal 2, we get 4a minus 2b plus 3.

03:06 What's the limit of the function f of x when x approaches 2? when x approaches 2 from the left, well, we have to take into account the first function is limit when x approaches 2 from the left, then the function.

03:30 To be considered is x squared minus 4 over x minus 2 because x is less than 2 if it approaches 2 from the left and this function as a removable discontinuity we can rewrite the function if we factorized in numerator this is x minus 2 x plus 2 over x minus 2.

04:03 So you can see that x minus 2 can be cancelled if x is different from 2 and when we are calculating the limit the limit is just means just that x approaches 2 but it's not equal so if we cancel this factor we're just left with the function x plus 2, and this limit is just 4...

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