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Chapter 1: Problem 41

Graphing a Function. (a) use a graphing utility to graph the function and (b) state the domain and range of the function. $$s(x)=2\left(\frac{1}{4} x-\left[\frac{1}{4} x\right]\right)$$

Short Answer

Expert verified
The graph of the function appears as a set of parallel lines each having a slope of \(2\). The domain of the function is all real numbers \(-\infty < x < \infty\) and the range of the function is \[0 鈮� s(x) < 2\].

Step by step solution

01

Understanding the function

The function provided is a piecewise function with a fractional part function defined as \(s(x)=2\left(\frac{1}{4} x-\left[\frac{1}{4} x\right]\right)\), where \([x]\) signifies the greatest integer less than or equal to \(x\).
02

Graphing the function

While graphing, assume an interval for \(x\) and replace those values in the function to get corresponding \(y\) or \(s(x)\) values. With these set of points, the function can be sketched by plotting the points on a graph. However, as directed, use a graphing utility for this task. The graph will appear as a set of parallel lines each having a slope of \(2\). It provides a clear visualization of the behavior of the given function.
03

Determining the domain

The domain of a function is the set of all real values of \(x\) that will give real values for the function. The given function takes all real x-values and hence the domain is all real numbers, denoted as \(-\infty < x < \infty\).
04

Determining the range

The range of a function is the set of all possible values of the function. In the given function, \(s(x)\), because of the fractional part, the output \(y\) or \(s(x)\) values will lie between 0 (inclusive) and 2 (exclusive). Thus the range is \[0 鈮� s(x) < 2\].

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Key Concepts

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

Piecewise Functions
Piecewise functions are fascinating mathematical entities. They are composed of different sub-functions, each applying to a certain interval of the main function's domain. This means, for each range of input values, a different rule or formula is applied. Piecewise functions are very useful for modeling situations where different scenarios occur within different ranges of the variable. In real-life applications, think of how companies may charge different rates at different times of the day or how a speed limit may vary depending on the part of the road.
  • A piecewise function is written using a ":" to separate different conditions and formulas.
  • It can have various conditions like inequalities "<", "<=", ">", or ">=", indicating which formula/applications they control.
  • Interpreting these conditions is crucial when plotting or analyzing these functions as each section of the graph represents a different rule or behavior.
Fractional Part Function
The fractional part function is another intriguing mathematical concept. It is defined as the difference between a number and the greatest integer less than or equal to that number. Simply put, for any real number "x", the fractional part "\(\{x\}\)" is given by "\(x - \lfloor x \rfloor\)". This function is important in various fields, including computer science and engineering, especially in numerical computations where precision needs to be managed.
  • It extracts the "decimal" part of any real number.
  • For example, if \(x = 4.7\), then the integer part is 4, and the fractional part function would return \(0.7\).
  • Graphically, the fractional part function resembles a sawtooth wave, repeating every integer step.
Domain and Range
Understanding domain and range is crucial when dealing with functions. These two properties give us a qualified understanding of the input-output behavior of a function. The **domain** of a function is all the possible input values (x-values) that the function can accept without causing any undefined behavior. For many functions, particularly polynomial functions, the domain can be all real numbers. However, piecewise functions or those involving fractions, square roots, or logarithms may have restricted domains due to the mathematical constraints those operations impose. On the other hand, the **range** is the set of all possible output values (y-values) the function can produce.
  • To determine the domain, consider any restrictions on x from the function鈥檚 definition.
  • For finding the range, observe the output possibilities which the function鈥檚 formula can achieve.
  • Graphing the function can often visually represent both domain and range, highlighting how x-values (domain) transform into y-values (range).

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Most popular questions from this chapter

Parallel and Perpendicular Lines, determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l}{L_{1} :(3,6),(-6,0)} \\ {L_{2} :(0,-1),\left(5, \frac{7}{3}\right)}\end{array}$$

Intercept Form of the Equation of a line, use the intercept form to find the equation of the line with the given intercepts. The intercept form of the equation of a line with intercepts \((a, 0)\) and \((0, b)\) is $$\frac{x}{a}+\frac{y}{b}=1, a \neq 0, b \neq 0$$ $$\begin{array}{l}{x \text { -intercept: }(2,0)} \\ {y \text { -intercept: }(0,3)}\end{array}$$

Evaluating a Difference Quotient In Exerrisess \(77-84,\) find the difference quotient and simplify your answer. $$ f(x)=5 x-x^{2}, \quad \frac{f(5+h)-f(5)}{h}, \quad h \neq 0 $$

The number of lumens (time rate of flow of light) \(L\) from a fluorescent lamp can be approximated by the model $$L=-0.294 x^{2}+97.744 x-664.875, \quad 20 \leq x \leq 90$$ where \(x\) is the wattage of the lamp. (a) Use a graphing utility to graph the function. (b) Use the graph from part (a) to estimate the wattage necessary to obtain 2000 lumens.

Data Analysis: Light lntensity A light probe is located \(x\) centimeters from a light source, and the intensity \(y\) (in microwatts per square centimeter) of the light is measured. The results are shown as ordered pairs \((x, y) .\) (Spreadsheet at LarsonPrecalculus.com) A model for the data is \(y=262.76 / x^{2.12}\) . (a) Use a graphing utility to plot the data points and the model in the same viewing window. (b) Use the model to approximate the light intensity 25 centimeters from the light source.
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