Finding the value of k for which the Piecewise Function is Continuous and Computing ff(5)

Описание: After watching this video, you would be able to find the value of k for which the Piecewise Function is continuous, and compute the numerical value of ff(5) from the piecewise function.

A piecewise function is a function defined by different expressions or rules for different intervals of its domain. It's like a chameleon that changes its colour (formula) depending on the input value(domain).

Here's a simple example:
f(x) = x^2 if x is less than 0
or
f(x) = x + 1 if x greater than or equal to 0

In this case, the function behaves differently based on whether x is negative or non-negative.

Piecewise functions are useful for modeling real-world situations where the behavior changes abruptly, like:

Tax brackets (different tax rates for different income levels)
Shipping costs (different rates for different weight ranges)
Electrical circuits (different behaviors for different voltage ranges)
Continuity of a Piecewise Function
A piecewise function is said to be continuous if it satisfies the following conditions:

Conditions for Continuity:
1. *Each Piece is Continuous*: Each sub-function is continuous within its respective interval.
2. *Continuity at Breakpoints*: The function is continuous at the points where the definition changes (breakpoints). This means:
The left-hand limit and right-hand limit at the breakpoint are equal.
The limit at the breakpoint equals the function value at that point.

Formally:
For a piecewise function \( f(x) \) defined as:
[ f(x) = \begin{cases}
f_1(x) & \text{if } x less than a \
f_2(x) & \text{if } x \geq a
\end{cases} ]

The function is continuous at ( x = a ) if:
1. \( \lim_{x \to a^-} f(x) = lim_{x \to a^+} f(x) \)
2. \( \lim_{x \to a} f(x) = f(a) \)

Example:
Consider the piecewise function:
f(x) = \begin{cases}
x^2 & \text{if } x less than 2 \
2x & \text{if } x \geq 2
\end{cases}

To check continuity at \( x = 2 \):
1. *Left-Hand Limit*: ( lim_{x \to 2^-} x^2 = 4 \)
2. *Right-Hand Limit*: ( \lim_{x \to 2^+} 2x = 4 \)
3. *Function Value*: ( f(2) = 2(2) = 4 )

Since the left-hand limit, right-hand limit, and function value are all equal, the function is continuous at ( x = 2 ).

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