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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 33
Chapter 4, Problem 33

Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = (x2 + 12)/(2x + 1)

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Identify the domain of the function f(x) = (x^2 + 12)/(2x + 1). The denominator cannot be zero, so set 2x + 1 ≠ 0 and solve for x to find the domain.
Find the x-intercepts by setting the numerator equal to zero: x^2 + 12 = 0. Solve for x to determine if there are any real x-intercepts.
Determine the y-intercept by evaluating f(0). Substitute x = 0 into the function and simplify to find the y-intercept.
Identify any vertical asymptotes by setting the denominator equal to zero: 2x + 1 = 0. Solve for x to find the vertical asymptote.
Analyze the end behavior of the function by finding the horizontal asymptote. Compare the degrees of the numerator and denominator. Since the degree of the numerator is greater than the degree of the denominator, perform polynomial long division to find the slant asymptote.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Analysis

Function analysis involves examining the properties of a function, such as its domain, range, intercepts, and asymptotes. For the given function f(x) = (x² + 12)/(2x + 1), understanding these properties is crucial for graphing. This includes identifying where the function is defined and any points where it may not exist, such as vertical asymptotes.
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Limits and Asymptotes

Limits help determine the behavior of a function as it approaches certain points, particularly at infinity or points of discontinuity. For rational functions like f(x), vertical asymptotes occur where the denominator equals zero, while horizontal asymptotes indicate the function's behavior as x approaches infinity. Analyzing these limits is essential for sketching the graph accurately.
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Critical Points and Derivatives

Critical points are values of x where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points for f(x), one must compute the derivative and solve for x. Understanding the behavior of the function at these points is vital for creating a complete and accurate graph.
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