Graph each rational function. ƒ(x)=4/(x-1)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{4}{x - 1}\). Notice that the denominator \(x - 1\) cannot be zero, so \(x \neq 1\).
2. Determine the vertical asymptote by setting the denominator equal to zero: solve \(x - 1 = 0\), which gives \(x = 1\). This means the graph will have a vertical asymptote at \(x = 1\).
3. Find the horizontal asymptote by analyzing the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity. Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is \(y = 0\).
4. Calculate a few key points by substituting values of \(x\) into the function, such as \(x = 0\), \(x = 2\), and \(x = -1\), to understand the shape and position of the graph relative to the asymptotes.
5. Sketch the graph using the vertical asymptote at \(x = 1\), the horizontal asymptote at \(y = 0\), and the plotted points. Remember the graph will approach but never touch the asymptotes.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined points. Graphing involves analyzing asymptotes, intercepts, and behavior near discontinuities.

Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function equals zero, causing the function to approach infinity or negative infinity. For f(x) = 4/(x-1), the vertical asymptote is at x = 1, indicating the graph will never cross this line but will approach it closely.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches infinity or negative infinity. For f(x) = 4/(x-1), since the degree of the numerator is less than the denominator, the horizontal asymptote is y = 0, meaning the graph approaches the x-axis at extreme values.