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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 2.14a
Capitolo 2, Problema 2.14a

Let g(x)=x3−4x8∣x−2∣g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\). <IMAGE>
Calculate g(x)g\(\left\)(x\(\right\)) for each value of xx in the following table.

Guida verificata passo dopo passo
1
Identify the function given: \( g(x) = \frac{x^3 - 4x}{8|x-2|} \). This function involves a polynomial in the numerator and an absolute value in the denominator.
Understand the absolute value function: \(|x-2|\) means that the expression inside the absolute value will be positive regardless of the sign of \(x-2\). This affects the domain and behavior of the function.
For each value of \(x\) provided in the table, substitute \(x\) into the function \(g(x)\). This involves calculating \(x^3 - 4x\) for the numerator and \(8|x-2|\) for the denominator.
Simplify the expression for each \(x\) by performing the arithmetic operations: calculate the cube and linear terms in the numerator, and evaluate the absolute value in the denominator.
Ensure that the denominator is not zero for any \(x\) value, as this would make the function undefined. If \(x = 2\), the denominator becomes zero, so check for this condition and handle it appropriately.

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Concetti chiave

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Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, evaluating g(x) means replacing x in the expression g(x) = (x^3 - 4x) / (8|x - 2|) with given values of x to find corresponding outputs. Understanding how to perform this substitution is crucial for calculating the function's values.
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Evaluating Composed Functions

Absolute Value

The absolute value of a number represents its distance from zero on the number line, disregarding its sign. In the function g(x), the term |x - 2| indicates that the output will depend on whether x is greater than or less than 2. This concept is essential for correctly interpreting the function's behavior and ensuring accurate calculations.
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Average Value of a Function

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The numerator of g(x), which is x^3 - 4x, is a polynomial of degree three. Understanding polynomial behavior, such as roots and end behavior, is important for analyzing the function's overall characteristics and its graph.
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Taylor Polynomials
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