Question

Sketch the graph of $f$ $$f(x)=\frac{-3}{x+3}$$

   Sketch the graph of $f$
$$f(x)=\frac{-3}{x+3}$$

Precalculus: Functions and Graphs

Precalculus: Functions and Graphs

Earl W. Swokowski,… 11th Edition

Chapter 3, Problem 8

Instant Answer

Solved by Expert Sriram Soundarrajan

10/27/2020

Step 1

The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. In this case, the function $f(x)=\frac{-3}{x+3}$ is defined for all real numbers except $x=-3$. This is because if Show more…

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Sketch the graph of $f$ $$f(x)=\frac{-3}{x+3}$$

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Key Concepts

Vertical Asymptote

A vertical asymptote occurs where the function is undefined due to the denominator equaling zero. This concept is crucial in graphing rational functions because these are the x-values that the function cannot take, and near which the function values grow without bound, either in the positive or negative direction.

Horizontal Asymptote

The horizontal asymptote describes the value that the function approaches as x tends to positive or negative infinity. It is determined by comparing the degrees of the polynomials in the numerator and denominator, and indicates the end-behavior of the rational function.

Rational Functions

Rational functions are ratios of polynomials and can exhibit behaviors such as asymptotes, holes, and intercepts. The overall shape of their graphs is determined by the degrees of the numerator and denominator, as well as the coefficients, which dictate the function's behavior for large values of x and near points where the function is undefined.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a base function to obtain the graph of a more complex function. Understanding these transformations allows one to quickly sketch the graph of functions by applying the corresponding movements and scaling factors derived from algebraic modifications.

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