For what values of a and b is f(x)={ -2, x ≤-1 a x-b, -1<x<1 3, x ≥ 1

step 1 So we have this one. This one is going to still follow the same thing. If you look here, you can

For what values of a and b is f(x)={ -2, x ≤-1 ...

Key Concepts

Piecewise-Defined Functions

A piecewise-defined function is one that is described by different expressions over different parts of its domain. Each segment of the function is defined on a specific interval. Analyzing such functions often requires checking the behavior at the points where the definition changes, as these are potential locations for discontinuities.

Matching Boundary Conditions

For piecewise-defined functions, ensuring overall continuity involves matching the limits from the left and right at the boundaries where the function’s definition changes. This often translates into setting up equations that equate the function’s limiting values at the transition points, thereby aligning the pieces to form a continuous function.

Continuity

Continuity is a fundamental concept in calculus which requires that a function does not have any abrupt changes or breaks. For a function to be continuous at a point, the limit of the function as the input approaches that point must equal the function’s value at that point. This concept ensures the function behaves predictably and smoothly.

Transcript

00:01 So we have this one.

00:04 This one is going to still follow the same thing.

00:07 If you look here, you can see that the bone of contention or the numbers and the consideration is negative one.

00:13 That is where the function is breaking.

00:16 It's also breaking at 1, right? so you have negative 1 and 1.

00:18 That is where you can have potential or discontinuities, right? so that is where our limit point is going to be, right? so we need to find the limit as x approaches negative 1.

00:31 And then the limit as x approaches positive one right and we know that for the limit as as as approaches negative one to exist then we've got to have the left -hand limit and the right -hand limit for the negative one to be true to be the same and then if the limit as this one also approaches positive one to exist we have to have the left -hand limit and the right -hand limit has the same right so we're going to apply the same thing same concept here so for, let's take the negative one, the first one.

01:05 So this first one, let's take the left hand limit.

01:10 The limit as x approaches negative one from the left of the function f of x.

01:16 What is happening? as we're approaching negative one from a smaller value, you can see that x is less than or equal to negative one.

01:24 So we're approaching negative one from a smaller value here.

01:27 Then a functional value or the function is going to be negative two, right? so we have our negative 2 here.

01:34 On the right hand limit, the limit as s approaches negative 1 from the right.

01:40 So we're approaching negative 1 from a bigger value, right? and this is where it is because all the values here are bigger than negative 1, right? so of f of f of x, what we're going to get here is a times negative 1.

01:57 We're going to start in negative 1 into the equation.

02:00 Using this equation, the second one, right? and so this is going to be b.

02:05 So this is negative a minus b, right? that is for the left hand limit.

02:10 Now we go to the right hand limit, this one, right? that is for the, a bigger pardon, that is for the limit as x approaches negative one, right? now we go to limit as x approaches positive one...

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