Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x−1)

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 45

4장, 문제 45

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

검증된 단계별 안내

Identify the base function given, which is \(f(x) = \frac{1}{x}\). This is a rational function with a vertical asymptote at \(x=0\) and a horizontal asymptote at \(y=0\).

Recognize that the function \(g(x) = \frac{1}{x-1}\) is a horizontal shift of the base function \(f(x) = \frac{1}{x}\). The expression \(x-1\) inside the denominator indicates a shift to the right by 1 unit.

Determine the new vertical asymptote by setting the denominator equal to zero: \(x - 1 = 0\), which gives \(x = 1\). This means the vertical asymptote moves from \(x=0\) to \(x=1\).

Note that the horizontal asymptote remains unchanged at \(y=0\) because the degree of the numerator and denominator are the same and the leading coefficients are unchanged.

Sketch the graph by shifting the original graph of \(f(x) = \frac{1}{x}\) one unit to the right, keeping the shape of the hyperbola the same but with the vertical asymptote at \(x=1\) and the horizontal asymptote at \(y=0\).

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주요 개념

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Parent Rational Functions

The parent functions f(x) = 1/x and f(x) = 1/x² are basic rational functions with distinct shapes and asymptotes. Understanding their graphs, including vertical and horizontal asymptotes, helps in visualizing transformations applied to these functions.

Transformations of Functions

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For g(x) = 1/(x−1), the graph of f(x) = 1/x is shifted horizontally to the right by 1 unit, changing the location of asymptotes and key points.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal asymptotes describe end behavior. Identifying these helps in accurately sketching the graph of rational functions.