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A **piecewise function** is a function that is defined by different expressions based on distinct intervals of the input variable, usually denoted as x. In simpler terms, it's like a set of rules that apply under different conditions. For example, the function in our exercise is defined as follows:

• For values of x less than 1, the function is expressed as \(f(x) = -x^2\), which represents a downward-opening parabola.
• At exactly \(x = 1\), the function is defined as \(f(x) = 2\).
• For values of x greater than 1, the function becomes \(f(x) = x - 2\), a linear function described by a straight line.

Understanding piecewise functions is crucial since each interval may represent different real-world scenarios, and knowing which expression to apply is key to solving problems involving such functions.

The **left-hand limit** of a function as x approaches a value is the value that the function's outputs are getting closer to, specifically when approaching from the left side of that value. Mathematically, this is represented as \( \lim_{x \to c^-} f(x) \), where c is the point of interest.In the given exercise, to find the left-hand limit as \(x\) approaches 1, we must consider the expression valid for \(x < 1\). For our function, this part is defined by \(f(x) = -x^2\). As \(x\) gets closer to 1 from values less than 1, plugging these into the expression gives:

• \(f(x) = -x^2 \to -1\) as \(x \to 1^{-}\).

Thus, the left-hand limit of the function as \(x\) approaches 1 is \(-1\). It's essential in calculus since it helps in determining if a two-sided limit exists.

The **right-hand limit** is the value a function approaches as x comes closer to a number from the right side, or greater values. It is denoted by \( \lim_{x \to c^+} f(x) \).For the function in our exercise, to determine the right-hand limit as \(x\) nears 1, we need the expression for \(x > 1\), which is \(f(x) = x - 2\).By evaluating values approaching 1 from the right, we have:

• \(f(x) = x - 2 \to -1\) as \(x \to 1^{+}\).

This shows that the right-hand limit of the function when \(x\) approaches 1 is also \(-1\). Just like the left-hand limit, the right-hand limit is important for deciding whether a full limit exists for a piecewise function.

**Graphing functions**, especially piecewise functions, involves plotting each piece of the function's expression over the designated interval. It's vital as it visually represents how the function behaves in different scenarios and aids in understanding possible limits.When graphing a piecewise function like the one in our exercise:

• Begin by plotting \(f(x) = -x^2\) for \(x < 1\), making sure it resembles a downward-opening parabola stopping just before \(x = 1\).
• Next, plot the discrete point \((1, 2)\), since the function takes a unique value at \(x = 1\).
• Finally, plot the line \(f(x) = x - 2\) for \(x > 1\), starting from a point just right of \((1, -1)\) and extending outward.

Accurate graphing helps in calculating limits by allowing you to visually inspect the behavior of the function as \(x\) approaches specific points, reinforcing the numerical process with a graphical one. This insight is particularly useful when determining whether one-sided or overall limits exist.