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Chapter 2: Problem 27

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=-|x| ;(-\infty, 0)$$

Short Answer

Expert verified
Over the interval \((- fty, 0)\), the function is increasing.

Step by step solution

01

Understand the Function

The function given is \( f(x) = -|x| \). Since \(|x|\) represents the absolute value of \(x\), \( f(x) \) will flip every positive output of \( |x| \) to its negative, making the graph of \( f(x) \) an inverted "V" shape.
02

Identify the Interval

The interval we are considering is \((-fty, 0)\), which means we are examining the behavior of the function only for negative values of \(x\).
03

Analyze Function Behavior

For negative values of \(x\), \(|x| = -x\). This means \( f(x) = -(-x) = x \). Therefore, within the interval \((-fty, 0)\), the function behaves like \(x\) which implies it's a linear function with a positive slope.
04

Observe Graph Behavior

Since \( f(x) = x \) within our interval \((-fty, 0)\), as we move from left to right (approaching zero from the negative side), the function is increasing.
05

Fill in the Blank

Over the interval \((-fty, 0)\), the function \( f(x) = -|x| \) is increasing as it is equivalent to \( f(x) = x\) in this interval.

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by the symbol \(|x|\). This makes \(|x|\) always non-negative. For example, \(|-3| = 3\) and \(|3| = 3\). In essence, it strips the number of its sign.

In the context of the function \(f(x) = -|x|\), the absolute value takes \(x\) and converts any negative values to positive before the negative sign in front of \(f(x)\) reverses it again. This characteristic gives the graph a signature inverted 'V' shape. Understanding absolute value is key for predicting how the \(f(x) = -|x|\) will transform across different intervals.
Interval Notation
Interval notation is a concise way of describing a range of values along a number line. It uses brackets or parentheses to signify whether endpoints are included or excluded.
  • Parentheses \( ( ) \) denote that an endpoint is not included.
  • Brackets \( [ ] \) indicate that an endpoint is included.
In the problem, the interval \((-\infty, 0)\) signifies that we are looking at all numbers less than zero, excluding zero itself.

Understanding interval notation helps in demarcating which portion of the function's behavior you're interested in analyzing. It鈥檚 particularly useful for identifying and describing segments of a function's graph within certain bounds.
Function Behavior
Function behavior describes how a function acts over a set range of inputs鈥攚hether it increases or decreases. When considering behavior, especially within a given interval, you focus on how the function's outputs change as you progress from one end of the interval to the other.

For the function \(f(x) = -|x|\) over the interval \((-\infty, 0)\), you see that \(|x| = -x\) for negative \(x\). Therefore, \(f(x) = -(-x) = x\), making not \(f(x)\) a straightforwardly linear function with a positive slope in this interval. Thus, it increases as you move towards zero from negative infinity.
Graph Analysis
Graph analysis involves interpreting the visual behavior of a function. You're examining how the function's graph moves, rotates, and scales across a coordinate plane.

For \(f(x) = -|x|\), the graph typically forms an inverted 'V'. However, considering the specified interval \((-\infty, 0)\), it translates into a rising line from left to right for negative \(x\) values.
  • As \(x\) approaches zero, the graph of this function rises, indicating an increase.
  • Graph analysis simplifies the understanding of complex functions by visualizing their behavior.
By visualizing the function \(f(x) = -|x|\), you quickly see the increasing pattern over the interval provided.

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Most popular questions from this chapter

The area \(\mathscr{A}\) of an equilateral triangle with sides of length \(x\) is given by $$\mathscr{A}(x)=\frac{\sqrt{3}}{4} x^{2}$$ (a) Find \(\mathscr{A}(2 x),\) the function representing the area of an equilateral triangle with sides of length twice the original length. (b) Find analytically the area of an equilateral triangle with side length 16. Use the given formula for \(\mathscr{A}(x)\) (c) Support the result of part (b) graphically.

Functions such as the pairs in Exercises \(69-72\) are called inverse functions, because the result of composition in both directions is the identity function. (Inverse functions will be discussed in detail in Section 5.1.) In a square viewing window, graph \(y_{1}=\sqrt[3]{x-6}\) and \(y_{2}=x^{3}+6,\) an example of a pair of inverse functions. Now graph \(y_{3}=x .\) Describe how the graph of \(y_{2}\) can be obtained from the graph of \(y_{1},\) using the graph \(y_{3}=x\) as a basis for your description.

A common air pollutant responsible for acid rain is sulfur dioxide \(\left(\mathrm{SO}_{2}\right) .\) Emissions of \(\mathrm{SO}_{2}\) during year \(x\) are computed by \(f(x)\) in the table. Emissions of carbon monoxide (CO) are computed by \(g(x)\) Amounts are given in millions of tons. $$\begin{array}{c|c|c|c|c|c}\boldsymbol{x} & 1970 & 1980 & 1990 & 2000 & 2010 \\\\\hline \boldsymbol{f}(\boldsymbol{x}) & 31.2 & 25.9 & 23.1 & 16.3 & 13.0 \\\\\hline \boldsymbol{g}(\boldsymbol{x}) & 204.0 &185.4 & 154.2 & 114.5 & 74.3\end{array}$$ (a) Evaluate \((f+g)(2010)\) (b) Interpret \((f+g)(x)\) (c) Make a table for \((f+g)(x)\)

Suppose that the length of a rectangle is twice its width. Let \(x\) represent the width of the rectangle. (a) Write a formula for the perimeter \(P\) of the rectangle in terms of \(x\) alone. Then use \(P(x)\) notation to describe it as a function. What type of function is this? (b) Graph the function \(P\) found in part (a) in the window \([0,10]\) by \([0,100] .\) Locate the point for which \(x=4,\) and explain what \(x\) represents and what \(y\) represents. (c) On the graph of \(P\), locate the point with \(x\) -value 4 Then sketch a rectangle satisfying the conditions described earlier, and evaluate its perimeter if its width is this \(x\) -value. Use the standard perimeter formula. How does the result compare with the \(y\) -value shown on your screen? (d) On the graph of \(P\), find a point with an integer \(y\) -value. Interpret the \(x\) - and \(y\) -coordinates here.

For each pair of functions, (a) find \((f+g)(x),(f-g)(x),\) and \((f g)(x)\) (b) give the domains of the functions in part (a); (c) find \(\frac{f}{8}\) and give its domain; (d) find \(f \circ g\) and give its domain; and (e) find \(g \circ f\) and give its domain. Do not use a calculator. $$f(x)=\sqrt[3]{x+4}, g(x)=x^{3}+5$$
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