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Chapter 0: Problem 33

\(f(x)=\frac{|x|}{x}\) (a) \(f(2)\) (b) \(f(-2)\) (c) \(f(x-1)\)

Short Answer

Expert verified
The solution for the given function: \(f(2)=1\), \(f(-2)=-1\), and \(f(x-1)=\frac{|x-1|}{x-1}\)

Step by step solution

01

Substitute value 2

Substitute \(x\) with 2 in \(f(x)=\frac{|x|}{x}\): \(f(2)=\frac{|2|}{2}\). We calculate the absolute value of 2 which remains 2, thus the function simplifies to \(f(2)=\frac{2}{2}=1\)
02

Substitute value -2

Next, substitute \(x\) with -2 in \(f(x)=\frac{|x|}{x}\): \(f(-2)=\frac{|-2|}{-2}\). The absolute value of -2 is 2, so the function simplifies to \(f(-2)=\frac{2}{-2}=-1\).
03

Substitute expression \(x-1\)

Lastly, we need to substitute \(x-1\) for \(x\) in the initial function: \(f(x-1)=\frac{|x-1|}{x-1}\). This is as far as we can simplify without specific values for \(x\). As we've noticed, depending on whether \(x-1\) is positive or negative we will obtain different results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
The concept of absolute value is fundamental in mathematics, often introduced in the context of real numbers. The absolute value of a number, represented as \(|x|\), refers to its distance from zero on the number line without considering its direction. This means:
  • For any positive number, such as 2, the absolute value remains the same. Therefore, \(|2| = 2\).
  • For negative numbers, like -2, the absolute value becomes positive, so \(|-2| = 2\).
By focusing on distance alone, absolute values are pivotal in equations that involve symmetry. In the function \(f(x)=\frac{|x|}{x}\), the absolute value determines whether the result of the function turns positive or negative by normalizing the numerator to be always positive. This concept allows for easy exploration of how the function \(f(x)\) behaves for both positive and negative values of \(x\).
Exploring Function Substitution
Function substitution involves replacing variables in a mathematical function with specific values or expressions. This technique simplifies complex expressions and allows for the exploration of new patterns and solutions.
  • In part (a) of the exercise, substituting \(x\) with 2 leads us to \(f(2)=\frac{|2|}{2}=1\). Here, using substitution, we see how the function behaves with the positive number 2.
  • In part (b), by substituting \(x\) with -2, we observe \(f(-2)=\frac{|-2|}{-2}=-1\). The function value turns negative, depicting the change due to absolute value manipulation.
  • Part (c) involves substituting \(x\) with an expression, resulting in \(f(x-1)=\frac{|x-1|}{x-1}\). Without specific numbers, the direct simplification isn鈥檛 feasible, indicating dependence on the sign of \(x-1\).
Function substitution proves to be a versatile tool, especially in testing how functions react to different inputs and expressions. It gives insights into both simple value inputs and more complex expressions.
Insights into Trigonometry Problems
While piecewise functions like \(f(x)=\frac{|x|}{x}\) don't directly involve trigonometry, understanding such functions build a foundation that aids in tackling trigonometric problems later. Here are a few ways this understanding is applicable:
  • Trigonometric identities often require manipulation of absolute values, especially in periodic functions where symmetry around the axes is crucial.
  • Substitution, a skill honed in this exercise, is essential in trigonometry for simplifying complex formulas and solving equations.
  • Piecewise functions reflect concepts found in trigonometric graphs like sine and cosine that have similar behaviors at intervals, where knowing how absolute value impacts the graph is helpful.
By studying piecewise and absolute value functions, students can develop problem-solving skills and strategies applicable to more advanced trigonometric concepts in calculus and analytic geometry.

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Most popular questions from this chapter

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 80 feet. $$ t_{1}=1, t_{2}=2 $$

Maximum Profit The cost per unit in the production of a portable CD player is \(\$ 60\). The manufacturer charges \(\$ 90\) per unit for orders of 100 or less. To encourage large orders, the manufacturer reduces the charge by \(\$ 0.15\) per CD player for each unit ordered in excess of 100 (for example, there would be a charge of \(\$ 87\) per CD player for an order size of 120 ). (a) The table shows the profit \(P\) (in dollars) for various numbers of units ordered, \(x\). Use the table to estimate the maximum profit. \begin{tabular}{|l|c|c|c|c|} \hline Units, \(x\) & 110 & 120 & 130 & 140 \\ \hline Profit, \(P\) & 3135 & 3240 & 3315 & 3360 \\ \hline \end{tabular} \begin{tabular}{|l|c|c|c|} \hline Units, \(x\) & 150 & 160 & 170 \\ \hline Profit, \(P\) & 3375 & 3360 & 3315 \\ \hline \end{tabular} (b) Plot the points \((x, P)\) from the table in part (a). Does the relation defined by the ordered pairs represent \(P\) as a function of \(x\) ? (c) If \(P\) is a function of \(x\), write the function and determine its domain.

Determine whether the function is even, odd, or neither. Then describe the symmetry. $$ f(x)=x \sqrt{1-x^{2}} $$

In Exercises 39-54, (a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=\sqrt[3]{x-1} $$

Your wage is \(\$ 8.00\) per hour plus \(\$ 0.75\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced is $$ y=8+0.75 x $$ (a) Find the inverse function. (b) What does each variable represent in the inverse function? (c) Determine the number of units produced when your hourly wage is \(\$ 22.25\).
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