Make a hand-drawn graph of each of the following. Check your...

Make a hand-drawn graph of each of the following. Check your results using a graphing calculator.
$$f(x)=\left\{\begin{array}{ll}\frac{x^{2}+3 x+2}{x+1}, & \text { for } x \neq-1 \\7, & \text { for } x=-1\end{array}\right.$$

Key Concepts

Removable Discontinuity

A removable discontinuity occurs in a function when a point is 'undefined' due to a factor that is common in both the numerator and the denominator and cancels out, creating a gap or hole in the graph. Although the simplified function appears continuous over that point, the original function’s definition and domain restrictions lead to a discontinuity that can be 'removed' by redefining the function's value at that point.

Graphical Analysis and Visualization

Graphing such functions involves illustrating the continuous part of the function while clearly marking any discontinuities, such as holes or jumps. For a rational function with a removable discontinuity, you typically draw the smooth curve for the simplified function and indicate the hole at the point of discontinuity. Additionally, if the function is defined differently at the discontinuity, such as assigning a specific value, that point should be distinctly plotted to reflect the function's complete behavior.

Piecewise Functions

A piecewise function is defined by different expressions over different parts of its domain. In this context, the function uses one expression when x is not equal to a particular value and a distinct value when x equals that specific number. Understanding piecewise functions is essential because it highlights how functions can change behavior at certain points or intervals and require careful consideration during analysis and graphing.

Rational Function Simplification

Simplifying a rational function often involves factoring its numerator and denominator to cancel common factors, while being mindful of restrictions in the domain. In the given example, the quadratic numerator can be factored, and a common factor with the denominator can be cancelled. However, the original restriction on the denominator remains, emphasizing that the simplified function is valid only over parts of the domain.

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