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x² +3x -4 g(x) = %3D x+1

x² +3x -4 g(x) = %3D x+1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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The given expression is: \[ g(x) = \frac{x^2 + 3x - 4}{x + 1} \] This equation represents a rational function, where the numerator is a quadratic polynomial \(x^2 + 3x - 4\) and the denominator is a linear polynomial \(x + 1\). Key features to consider for this function: 1. **Domain**: The function is undefined where the denominator equals zero. Thus, the domain excludes \(x = -1\). 2. **Simplification**: You can factor the quadratic numerator to see if it shares any factors with the denominator, allowing for simplification. 3. **Discontinuities**: Since \(x = -1\) causes the denominator to be zero, there may be a vertical asymptote at this point, unless the factor can be canceled. 4. **Behavior at Infinity**: The degree of the numerator and the denominator will determine the horizontal asymptote. Graphical representations or further analysis would typically include finding intercepts, asymptotes, and behavior at significant points to understand the complete behavior of the function.
Transcribed Image Text:The given expression is: \[ g(x) = \frac{x^2 + 3x - 4}{x + 1} \] This equation represents a rational function, where the numerator is a quadratic polynomial \(x^2 + 3x - 4\) and the denominator is a linear polynomial \(x + 1\). Key features to consider for this function: 1. **Domain**: The function is undefined where the denominator equals zero. Thus, the domain excludes \(x = -1\). 2. **Simplification**: You can factor the quadratic numerator to see if it shares any factors with the denominator, allowing for simplification. 3. **Discontinuities**: Since \(x = -1\) causes the denominator to be zero, there may be a vertical asymptote at this point, unless the factor can be canceled. 4. **Behavior at Infinity**: The degree of the numerator and the denominator will determine the horizontal asymptote. Graphical representations or further analysis would typically include finding intercepts, asymptotes, and behavior at significant points to understand the complete behavior of the function.
### Asymptotes and Additional Points **C) Find & Sketch the Vertical Asymptotes:** Vertical asymptotes occur where a function approaches infinity as the input approaches a certain value. To find vertical asymptotes, look for values of \( x \) that make the function's denominator zero (assuming it cannot be simplified further). **D) Find & Sketch Horizontal/Slant Asymptotes:** Horizontal asymptotes occur when a function approaches a specific value as \( x \) approaches infinity or negative infinity. Slant asymptotes may occur when there is a linear asymptote rather than horizontal. Examine the degrees of the polynomial or use long division to find the equation of slant asymptotes. **E) Find Additional Points Between and Beyond X-intercepts and Vertical Asymptotes:** To thoroughly understand the behavior of the function, identify additional points by substituting \( x \) values between and beyond the x-intercepts and vertical asymptotes. This can help in sketching an accurate graph. **Graph Explanation:** The graph shows a typical number line. The marking includes various intervals and specific points relevant for understanding the function's behavior across different segments of the line. These can be used to identify critical points including asymptotic behavior.
Transcribed Image Text:### Asymptotes and Additional Points **C) Find & Sketch the Vertical Asymptotes:** Vertical asymptotes occur where a function approaches infinity as the input approaches a certain value. To find vertical asymptotes, look for values of \( x \) that make the function's denominator zero (assuming it cannot be simplified further). **D) Find & Sketch Horizontal/Slant Asymptotes:** Horizontal asymptotes occur when a function approaches a specific value as \( x \) approaches infinity or negative infinity. Slant asymptotes may occur when there is a linear asymptote rather than horizontal. Examine the degrees of the polynomial or use long division to find the equation of slant asymptotes. **E) Find Additional Points Between and Beyond X-intercepts and Vertical Asymptotes:** To thoroughly understand the behavior of the function, identify additional points by substituting \( x \) values between and beyond the x-intercepts and vertical asymptotes. This can help in sketching an accurate graph. **Graph Explanation:** The graph shows a typical number line. The marking includes various intervals and specific points relevant for understanding the function's behavior across different segments of the line. These can be used to identify critical points including asymptotic behavior.
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