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A piecewise function is a special type of function that has different expressions or formulas for different parts of its domain. These functions look like a "piece" of different functions put together, based on certain rules. In this example, the function is:\[ g(x)=\begin{cases} \ x, & x<-2 \ b x^{2}, & x \geq-2 \end{cases} \]This means for any x less than \(-2\), the function will behave like the line \(x\), but for x at or greater than \(-2\), it behaves like \(b x^2\). These differences in behavior at specific parts of the domain often require special attention to ensure the entire function remains continuous.

The left-hand limit, often written as \(\lim_{{x \to c^-}} f(x)\), describes the value a function approaches as the input \(x\) gets closer to a certain point \(c\) from the left (or from smaller values). In our given piecewise function, we find the left-hand limit as x approaches -2:- We use the part of the function that applies to \(x < -2\), which is simply \(g(x) = x\).- As \(x\) approaches -2 from the left, \(g(x)\) approaches -2.Thus, the left-hand limit as x approaches -2 is \(-2\). This is a crucial step when checking for continuity in piecewise functions.

The right-hand limit \(\lim_{{x \to c^+}} f(x)\) shows the value a function approaches as the input \(x\) gets closer to the point \(c\) from the right (or from larger values). For our function \(g(x)\), we look at the right-hand limit as x approaches -2:- We use the part of the function that applies for \(x \geq -2\), which is \(g(x) = b x^2\).- As \(x\) nears -2 from the right, \(g(x) = b(-2)^2 = 4b\).Thus, the right-hand limit at x just greater than -2 is \(4b\). To ensure the piecewise function is continuous, this value must match the left-hand limit.

To achieve continuity at a point where a piecewise function changes its definition, we typically equate the left-hand and right-hand limits at that point. From our previous exploration:- The left-hand limit at \(x = -2\) is \(-2\).- The right-hand limit at \(x = -2\) is \(4b\).For the function \(g(x)\) to be continuous at this point, these limits must be equal. Therefore, we solve the equation:\[ -2 = 4b \]To solve for \(b\), we divide both sides by 4, yielding:\[ b = \frac{-2}{4} = -\frac{1}{2} \]This solution ensures that the function becomes continuous at \(x = -2\). The process illustrates the importance of solving equations when ensuring the continuity of piecewise functions.