Graph each of the following functions. Check your results...

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

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Graphing Techniques

Graphing techniques involve analyzing the overall behavior of a function, including identifying its domain, continuity, and points of discontinuity. Techniques such as identifying critical values, testing intervals, and using graphing calculators help in accurately sketching the graph of a function.

Removable Discontinuity

A removable discontinuity occurs when a function’s numerator and denominator share a common factor, which cancels out algebraically. This creates a 'hole' in the graph at a certain point even though the simplified expression appears continuous. Recognizing and properly handling removable discontinuities is essential when graphing rational functions.

Rational Functions

Rational functions are quotients of polynomials. Their graphs are characterized by typical behaviors such as asymptotes and discontinuities, which arise when the denominator equals zero. Understanding the algebraic structure of rational functions is crucial for determining their domain and graph features.

Piecewise Functions

Piecewise functions are defined by different expressions over different parts of their domain. They require careful analysis at points where the definition changes, as these points can lead to discontinuities or special conditions that affect the overall graph of the function.

Transcript

Please give Ace some feedback