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Chapter 1: Problem 15

\(f(x)=\frac{1}{x}\)

Short Answer

Expert verified
The graph of the function is a rectangular hyperbola. The domain and range of the function are all real numbers except zero.

Step by step solution

01

Identify the graph of the function

The graph of the function \(f(x)=\frac{1}{x}\) is a rectangular hyperbola. The graph for \(x>0\) lies entirely in the first quadrant and the graph for \(x<0\) lies entirely in the third quadrant. The function is undefined at \(x=0\).
02

Find the domain

The domain of the function \(f(x)=\frac{1}{x}\) is all the real numbers except for zero. That is because there is no real number such that substituting it into the function would give a result. So, the domain is \(x\neq0\).
03

Find the range

The function \(f(x)=\frac{1}{x}\) gives all the real numbers as outputs except for zero, as there is no real number such that substituting it into the function would give zero as the result. So, the range also is \(y\neq0\).

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Key Concepts

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

Domain and Range
Understanding the domain and range of functions is fundamental in calculus. For the function \(f(x) = \frac{1}{x}\), let's explore these concepts:
  • Domain: The domain of a function includes all possible input values (\(x\)) that produce real output values (\(f(x)\)). For \(f(x) = \frac{1}{x}\), you cannot substitute \(x = 0\) because dividing by zero is undefined. Therefore, the domain is all real numbers except zero, written as \(x eq 0\).
  • Range: The range consists of all the output values (\(y\)) that a function can produce. For \(f(x) = \frac{1}{x}\), as \(x\) approaches zero from both positive and negative sides, \(f(x)\) drastically increases or decreases without ever reaching zero. Thus, the range is all real numbers except zero, represented as \(y eq 0\).
These restrictions define the behavior and limitations of our function.
Graphing Functions
Graphing functions helps visualize the relationship between \(x\) and \(f(x)\). For \(f(x) = \frac{1}{x}\), its graph is known for its distinctive shape:- The graph can be dissected into two distinct sections: - For \(x > 0\), the curve is located in the first quadrant, where both \(x\) and \(f(x)\) are positive. - For \(x < 0\), the curve lies in the third quadrant, where both \(x\) and \(f(x)\) are negative.
- The graph never touches the axes, reflecting its undefined nature (like the domain and range restrictions). As \(x\) moves towards zero, \(f(x)\) approaches infinity.Understanding these principles aids in identifying functional behaviors and critical characteristics.
Rectangular Hyperbola
The function \(f(x) = \frac{1}{x}\) graphs as a rectangular hyperbola, an important concept in calculus. Here's what you need to know:
  • Symmetry: This type of function is mirrored across both the y-axis and the x-axis, demonstrating 180-degree rotational symmetry around the origin.
  • Asymptotes: The graph has asymptotes along the y-axis (\(x = 0\)) and the x-axis (\(y = 0\)), indicating the points it approaches but never reaches. This behavior reflects its undefined nature at specific values.
  • Quadrants: The function's path systematically covers the first and third quadrants, never crossing into the other quadrants due to the defined nature of \(f(x)\).
Rectangular hyperbolas are crucial in understanding the geometric aspects of functions, showcasing the interconnectedness between algebraic expressions and their graphical representations.

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Most popular questions from this chapter

Find the area under the curve \(y=2 x-x^{2}\) from \(x=1\) to \(x=2\) using the Midpoint Formula with \(n=4\) .

A solid is generated when the region in the first quadrant enclosed by the graph of \(y=\left(x^{2}+1\right)^{3},\) the line \(x=1,\) the \(x\) -axis, and the \(y\) -axis is revolved about the \(x\) axis. Its volume is found by evaluating which of the following integrals? (A) \(\pi \int_{1}^{8}\left(x^{2}+1\right)^{3} d x\) (B) \(\pi \int_{1}^{8}\left(x^{2}+1\right)^{6} d x\) (C) \(\pi \int_{0}^{1}\left(x^{2}+1\right)^{3} d x\) (D) \(\pi \int_{0}^{1}\left(x^{2}+1\right)^{6} d x\)

If \(\frac{d y}{d x}=\frac{e^{x}}{y^{2}}\) and \(y(0)=1,\) find an equation for \(y\) in terms of \(x\)

Use the method of cylindrical shells to find the volume of the solid that results when the region bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved around the \(x\) -axis.

Find the area of the region between the two curves in each problem, and be sure to sketch each one. (We gave you only endpoints in one of them) The answers are in Chapter 19 . The curve \(y=x^{2}\) and the curve \(y=4 x-x^{2}\).
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