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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.14a
Chapter 2, Problem 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.

Verified step by step guidance
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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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.
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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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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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Related Practice
Textbook Question

Complete the following steps for the given functions. 


a. Find the slant asymptote of ff.


f(x)=4x3+4x2+7x+4x2+1f\(\left\)(x\(\right\))=\(\frac{4x^3+4x^2+7x+4}{x^2+1}\)

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Textbook Question

For the following position functions, make a table of average velocities similar to those in Exercises 19–20 and make a conjecture about the instantaneous velocity at the indicated time. 


a. s(t)=−16t^2+80t+60 at t=3

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Textbook Question

Tangent lines with zero slope


a. Graph the function f(x)=x^2−4x+3.

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Textbook Question

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.


a. Graph the position function, for 0≤t≤9.

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The graph of a function can never cross one of its horizontal asymptotes.

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Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x2 − 9)/(x(x−3))

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