Textbook Question
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+x−6)/(x−3)
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 84
Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x4−3x2−4). If f(3)=−150, determine the value of a_n.
Verified step by step guidance1
Start with the given function: .
Substitute into the function to use the given value . This gives: .
Calculate the powers of 3 inside the parentheses: and .
Simplify the expression inside the parentheses: . Perform the multiplication and subtraction step-by-step.
After simplifying the parentheses, solve for by dividing both sides of the equation by the simplified value inside the parentheses.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Function Evaluation
Function evaluation involves substituting a specific input value into the function's expression to find the corresponding output. In this problem, you substitute x = 3 into f(x) to find f(3), which helps in solving for the unknown coefficient a_n.
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4:26
Evaluating Composed Functions
Polynomial Functions
Polynomial functions are expressions involving variables raised to whole-number exponents combined using addition, subtraction, and multiplication. Understanding the structure of the polynomial f(x) = a_n(x^4 − 3x^2 − 4) is essential to correctly substitute values and manipulate the equation.
Recommended video:
06:04Introduction to Polynomial Functions
Solving for an Unknown Coefficient
When a function includes an unknown coefficient, you can find its value by using given function values. Here, knowing f(3) = -150 allows you to set up an equation and solve for a_n by isolating it after substituting x = 3 into the polynomial.
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Guided course
6:54Cramer's Rule - 2 Equations with 2 Unknowns
Related Practice
Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. 2x2 + 5x - 3 ≤ 0
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Textbook Question
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+1)/x
604
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Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0
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Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. x2 + 3x - 10 > 0
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Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0
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