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Math
Calculus
g(x) = 2 x²_4 3x²+x-4

g(x) = 2 x²_4 3x²+x-4

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 image contains a series of layered mathematical expressions involving fractions and operations as follows: 1. The main expression is a division: \[ \frac{9}{6} \] 2. Underneath this, there is a fraction with the expression: \[ \frac{5 \frac{2}{3}}{4} \] 3. The entire setup suggests a complex fraction where one fraction is divided by another: \[ \frac{\frac{9}{6}}{\frac{5 \frac{2}{3}}{4}} \] 4. Understanding and solving this requires: - Simplifying each of the fractions as follows: - \(\frac{9}{6}\) simplifies to \(\frac{3}{2}\) after dividing by the greatest common divisor. - Convert \(5 \frac{2}{3}\) to an improper fraction: \(\frac{17}{3}\). - The division then looks like this: \[ \frac{\frac{3}{2}}{\frac{17}{3}} \] - Apply the rule for dividing fractions (multiply by the reciprocal): \[ \frac{3}{2} \times \frac{3}{17} = \frac{9}{34} \] This calculation showcases an application of dividing fractions and converting mixed numbers to improper fractions for simplification.
Transcribed Image Text:The image contains a series of layered mathematical expressions involving fractions and operations as follows: 1. The main expression is a division: \[ \frac{9}{6} \] 2. Underneath this, there is a fraction with the expression: \[ \frac{5 \frac{2}{3}}{4} \] 3. The entire setup suggests a complex fraction where one fraction is divided by another: \[ \frac{\frac{9}{6}}{\frac{5 \frac{2}{3}}{4}} \] 4. Understanding and solving this requires: - Simplifying each of the fractions as follows: - \(\frac{9}{6}\) simplifies to \(\frac{3}{2}\) after dividing by the greatest common divisor. - Convert \(5 \frac{2}{3}\) to an improper fraction: \(\frac{17}{3}\). - The division then looks like this: \[ \frac{\frac{3}{2}}{\frac{17}{3}} \] - Apply the rule for dividing fractions (multiply by the reciprocal): \[ \frac{3}{2} \times \frac{3}{17} = \frac{9}{34} \] This calculation showcases an application of dividing fractions and converting mixed numbers to improper fractions for simplification.
**Instructions for Sketching a Rational Function** When sketching a rational function, ensure to highlight the following key points and features: - **Hole**: Identify any point where the function is undefined due to a factor canceling out in both the numerator and denominator. - **Vertical Asymptote(s)**: Determine the values of \(x\) where the denominator equals zero, and there is no cancellation with the numerator, indicating an approach to infinity or negative infinity. - **Horizontal Asymptote or Slant Asymptote**: - **Horizontal Asymptote**: Analyze the degrees of the numerator and denominator. If they are equal, the horizontal asymptote is \(y =\) (leading coefficient of the numerator)/(leading coefficient of the denominator). If the degree of the numerator is less, the asymptote is \(y = 0\). - **Slant Asymptote**: Occurs when the degree of the numerator is exactly one greater than the degree of the denominator. Use polynomial division to find it. - **x-intercept(s)**: Set the numerator equal to zero and solve for \(x\). - **y-intercept(s)**: Calculate the value of the function when \(x = 0\).
Transcribed Image Text:**Instructions for Sketching a Rational Function** When sketching a rational function, ensure to highlight the following key points and features: - **Hole**: Identify any point where the function is undefined due to a factor canceling out in both the numerator and denominator. - **Vertical Asymptote(s)**: Determine the values of \(x\) where the denominator equals zero, and there is no cancellation with the numerator, indicating an approach to infinity or negative infinity. - **Horizontal Asymptote or Slant Asymptote**: - **Horizontal Asymptote**: Analyze the degrees of the numerator and denominator. If they are equal, the horizontal asymptote is \(y =\) (leading coefficient of the numerator)/(leading coefficient of the denominator). If the degree of the numerator is less, the asymptote is \(y = 0\). - **Slant Asymptote**: Occurs when the degree of the numerator is exactly one greater than the degree of the denominator. Use polynomial division to find it. - **x-intercept(s)**: Set the numerator equal to zero and solve for \(x\). - **y-intercept(s)**: Calculate the value of the function when \(x = 0\).
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