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Chapter 0: Problem 9

Find the natural domain and determine the range of each function. If you have a graphing utility, use it to confirm that your result is consistent with the graph produced by your graphing utility. [Note: Set your graphing utility in radian mode when $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{1}{x-3}} & {\text { (b) } F(x)=\frac{x}{|x|}} \\ {\text { (c) } g(x)=\sqrt{x^{2}-3}} & {\text { (d) } G(x)=\sqrt{x^{2}-2 x+5}} \\ {\text { (e) } h(x)=\frac{1}{1-\sin x}} & {\text { (f) } H(x)=\sqrt{\frac{x^{2}-4}{x-2}}}\end{array} $$ graphing trigonometric functions.]

Short Answer

Expert verified
Domains: \( f(x) \neq 3, F(x) \neq 0, g(x) \leq -\sqrt{3} \text{ or } \geq \sqrt{3}, G(x) \text{ all real, } h(x) \neq n\pi + \frac{\pi}{2}, H(x) \leq -2 \text{ or } \geq 2 \). Ranges: Real numbers except exceptions for each function as derived.

Step by step solution

01

Understanding the Function Domains

For the function \( f(x) = \frac{1}{x - 3} \), the domain is all real numbers except \( x = 3 \) because division by zero is undefined. For \( F(x) = \frac{x}{|x|} \), the domain is all real numbers except \( x = 0 \) due to the undefined division by zero when \( x = 0 \). For \( g(x) = \sqrt{x^2 - 3} \), \( x^2 - 3 \geq 0 \), so \( x \leq -\sqrt{3} \) or \( x \geq \sqrt{3} \). For \( G(x) = \sqrt{x^2 - 2x + 5} \), since the expression inside the square root \( x^2 - 2x + 5 \geq 0 \) yields true for all real \( x \), the domain is all real numbers. For \( h(x) = \frac{1}{1 - \sin x} \), it's all real numbers except when \( \sin x = 1 \), specifically \( x eq n\pi + \frac{\pi}{2} \) for integer \( n \). Lastly, \( H(x) = \sqrt{\frac{x^2 - 4}{x - 2}} \) is defined when \( x eq 2 \) and \( x^2 - 4 \geq 0 \), so \( x \leq -2 \) or \( x \geq 2 \).
02

Determining Function Ranges

For \( f(x) = \frac{1}{x - 3} \), the range is all real numbers except 0. For \( F(x) = \frac{x}{|x|} \), the range is \( \{-1, 1\} \) since it is determined by the sign of \( x \). For \( g(x) = \sqrt{x^2 - 3} \), the range is \([0, \infty)\) due to the non-negative square root. For \( G(x) = \sqrt{x^2 - 2x + 5} \), the range is \([\sqrt{3}, \infty)\) since the minimum value of the expression is 3. For \( h(x) = \frac{1}{1 - \sin x} \), the range covers \(( -\infty, -1) \cup (1, \infty)\) because \( 1 - \sin x \) varies between -1 and 1. For \( H(x) = \sqrt{\frac{x^2 - 4}{x - 2}} \), analyzing \( x \leq -2 \) or \( x \geq 2 \) shows the range is \([0, \infty)\).
03

Validate with Graphing Utility

Use a graphing utility in radian mode to verify the domain and range calculations for each function. Check that the graphical output matches each analytical derivation, confirming the absence of points where each function is undefined and the correct output range values.

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Key Concepts

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

Domain of a Function
The domain of a function consists of all the input values (often represented by \( x \)) for which the function is defined. Simply put, the domain tells us which values we can plug into our function without encountering mathematical problems like division by zero or taking roots of negative numbers.
For example, consider the function \( f(x) = \frac{1}{x-3} \). The denominator \( x-3 \) can never be zero, so \( x \) cannot be 3. Therefore, the domain of this function is all real numbers except \( x = 3 \).
Similarly, for functions involving square roots, like \( g(x) = \sqrt{x^2 - 3} \), we need the expression under the square root to be non-negative. Thus, \( x \) must be \( \leq -\sqrt{3} \) or \( x \geq \sqrt{3} \) to be in the domain.
  • Division by zero is undefined, avoid values that make the denominator zero.
  • For square roots, ensure the expression inside is zero or positive.
Range of a Function
The range of a function includes all possible output values, which the function can produce based on its domain. This is essential to understand what kind of results we might expect from the function after applying it to the elements within its domain.
For example, with \( f(x) = \frac{1}{x - 3} \), the function can output any real number except zero as the denominator can make \( f(x) \) large without actually reaching zero.
For \( F(x) = \frac{x}{|x|} \), which assigns \( -1 \) for negative \( x \) and \( 1 \) for positive \( x \), the outputs are filtered to only \( -1 \) and \( 1 \).
  • The range depends on calculated values as \( x \) varies across its domain.
  • Identify functions that have minimum or maximum values due to their form, e.g., square roots.
Graphing Utility
The use of a graphing utility helps to visually confirm our findings about the domain, range, and behavior of a function. It provides a graphical representation to better understand where certain points may be undefined or where the outputs are restricted.
To use a graphing utility effectively, ensure it's set to radian mode when working with functions involving angles or trigonometric calculations, as instructed. This ensures accurate representations when graphing functions like \( h(x) = \frac{1}{1 - \sin x} \), where understanding the periodic nature of sine is crucial.
Graphing utilities allow for:
  • Visual Verification: Graphs can reveal any discontinuities or asymptotic behavior.
  • Insights into Ranges: Seeing the outputs along the \( y \)-axis can confirm analytical range identifications.
Trigonometric Functions
Trigonometric functions such as sine, cosine, and tangent are foundational in analyzing functions involving angles or periodic phenomena. They have special properties and behavior that influence their domains and ranges.
For instance, the function \( h(x) = \frac{1}{1 - \sin x} \) is interesting because \( \sin x \) ranges between -1 and 1, but to maintain a valid denominator, we avoid \( \sin x = 1 \), leading to certain points being excluded from the domain.
When working with trigonometric functions:
  • The periodic nature means certain output values repeat regularly.
  • Different trigonometric functions have specific known ranges, e.g., \([-1, 1]\) for sine and cosine.
Understanding these can help predict the behavior of more complex functions that use them.

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Most popular questions from this chapter

Sketch the graph of a function that is one-to-one on \((-\infty,+\infty),\) yet not increasing on \((-\infty,+\infty)\) and not decreasing on \((-\infty,+\infty)\)

Use a calculating utility to approximate the expression. Round your answer to four decimal places. $$ \begin{array}{lll}{\text { (a) } \log 23.2} & {\text { (b) } \ln 0.74}\end{array} $$

An airplane is flying at a constant height of \(3000 \mathrm{ft}\) above water at a speed of \(400 \mathrm{ft} / \mathrm{s}\). The pilot is to release a survival package so that it lands in the water at a sighted point \(P .\) If air resistance is neglected, then the package will follow a parabolic trajectory whose equation relative to the coordinate system in the accompanying figure is $$ y=3000-\frac{g}{2 v^{2}} x^{2} $$ where \(g\) is the acceleration due to gravity and \(v\) is the speed of the airplane. Using \(g=32 \mathrm{ft} / \mathrm{s}^{2},\) find the "line of sight" angle \(\theta,\) to the nearest degree, that will result in the package hitting the target point.

Sketch the graph of the equation by making appropriate transformations to the graph of a basic power function. If you have a graphing utility, use it to check your work. $$ \begin{array}{ll}{\text { (a) } y=1-\sqrt{x+2}} & {\text { (b) } y=1-\sqrt[3]{x+2}} \\ {\text { (c) } y=\frac{5}{(1-x)^{3}}} & {\text { (d) } y=\frac{2}{(4+x)^{4}}}\end{array} $$

The number of hours of daylight on a given day at a given point on the Earth's surface depends on the latitude \(\lambda\) of the point, the angle \(\gamma\) through which the Earth has moved in its orbital plane during the time period from the vernal equinox (March \(21),\) and the angle of inclination \(\phi\) of the Earth's axis of rotation measured from ecliptic north \(\left(\phi \approx 23.45^{\circ}\right) .\) The number of hours of daylight \(h\) can be approximated by the formula $$ h=\left\\{\begin{array}{ll}{24,} & {D \geq 1} \\ {12+\frac{2}{15} \sin ^{-1} D,} & {|D|<1} \\ {0,} & {D \leq-1}\end{array}\right. $$ $$ \begin{array}{l}{\text { where }} \\ {\qquad D=\frac{\sin \phi \sin \gamma \tan \lambda}{\sqrt{1-\sin ^{2} \phi \sin ^{2} \gamma}}}\end{array} $$ and \(\sin ^{-1} D\) is in degree measure. Given that Fairbanks, Alaska, is located at a latitude of \(\lambda=65^{\circ} \mathrm{N}\) and also that \(\gamma=90^{\circ}\) on June 20 and \(\gamma=270^{\circ}\) on December \(20,\) approximate (a) the maximum number of daylight hours at Fairbanks to one decimal place (b) the minimum number of daylight hours at Fairbanks to one decimal place.
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