Find the values of a and b such that the function defined by...

Find the values of $a$ and $b$ such that the function defined by
$$f(x)=\left\{\begin{array}{ll}
5, & \text { if } x \leq 2 \\
a x+b, & \text { if } 2<x<10 \\
21, & \text { if } x \geq 10
\end{array}\right.$$
is a continuous function.

Key Concepts

Continuity

Continuity refers to the property of a function where it has no abrupt jumps, holes, or breaks at any point in its domain. A continuous function has the characteristic that the limit from the left equals the limit from the right and also equals the function's value at that point. This concept is fundamental in evaluating piecewise functions at the points where their definitions change.

Piecewise-defined Functions

Piecewise-defined functions are functions that are described by different expressions or rules over different intervals of the domain. Each segment of the function may have its own formula, and ensuring the overall function is continuous often requires careful handling of the points at which these segments meet.

Continuity Conditions at Boundaries

Continuity conditions at boundaries involve ensuring that the expressions defining a piecewise function yield the same value at the endpoints of the intervals. Specifically, to achieve continuity, jump discontinuities at the points where the function's definition changes must be eliminated by setting the left-hand and right-hand values equal to each other, which sometimes involves solving for unknown parameters.

Transcript

00:01 In this problem, we have been given a function f of x and we need to find the values of a and b so that the function is a continuous function.

00:07 Now, if the function is a continuous function, then that means that f of x is continuous for all real values of x.

00:14 So, at the point x equals to 2, f of x will be continuous.

00:19 Thus, limit x tends to 2 minus of fx is equal to limit x tends to 2 plus of fx is equal to f of 2.

00:28 This means that the left -hand limit is equal to the right -hand limit is equal to the value of the function at the point 2.

00:34 So the limit as x tends to 2 minus of fx will be the limit as x tends to 2 of 5 because fx is equal to 5 when x is less than 2.

00:45 This will be equal to the limit as extends to 2 of ax plus b because for x tends to 2 plus x will be greater than 2 and so fx will be a x plus b.

00:56 And this will be equal to f of 2, that will be equal to 5 because whenever x is less than are equal to 2, f of x is equal to 5.

01:05 So, since 5 is a constant, the limit extends to 2 of 5 will be 5.

01:11 We use the direct substitution property for the next one, so we have a times 2 plus b, and here we have 5.

01:18 So what we end up with is 2a plus b is equal to 5 and we name this equation 1.

01:25 Now, since the function is continuous for all real numbers, it will also be continuous at the point x equals to 10.

01:33 And hence the left -hand limit will be equal to the right -hand limit.

01:41 And this will be equal to the value of the function at x equals to 10.

01:46 So the limit has x -tenths to 10 minus of f -x.

01:50 Whenever x -tens to 10 minus x will be less than 10, so over here we will have a -x plus b...

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