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

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.14a

Chapter 2, Problem 2.14a

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

Verified step by step guidance

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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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 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.

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.

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

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Key Concepts

Function Evaluation

Absolute Value

Polynomial Functions

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