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Chapter 1: Problem 16

Determine the domain of the function and sketch the graph. $$g(x)=x+\frac{1}{x}$$.

Short Answer

Expert verified
The domain of the function \(g(x)=x+\frac{1}{x}\) is all real numbers except zero. The graph of the function has a vertical asymptote at x=0.

Step by step solution

01

Determine the Domain

The domain of a function consists of all the real values of x for which the function is defined. In this case, the domain is all real numbers except the number that makes the denominator of the fraction equal to zero. Hence, for the function \(g(x)=x+\frac{1}{x}\), the domain is given by \(D={x|x \neq 0}\). That is, all real numbers except zero.
02

Sketch the graph

To sketch the graph, create a table of values where x takes all real values except zero. By substituting these values into the function, obtain the corresponding y-values. After plotting these points on a graph, join them to form the curve. Note that as x approaches zero from the right, \(g(x)\) goes to infinity. As x approaches zero from the left, \(g(x)\) goes to negative infinity. Therefore, the function has a vertical asymptote at x=0.

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Key Concepts

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

Real Numbers
Real numbers are a fundamental concept in mathematics, encompassing almost all the numbers you encounter daily. They include integers, fractions, decimals, and irrational numbers. These are numbers like 0, -1, 2.5, or \(\sqrt{2}\), which can be plotted on an infinite line known as the number line.

When dealing with functions such as \(g(x) = x + \frac{1}{x}\), we typically explore the 'domain' of real numbers. The domain consists of all the input values (x-values) that allow the function to work without any issues.
  • Real numbers also exclude values that make the function undefined. For example, in a fraction, any value which makes the denominator zero is excluded, because division by zero is undefined.
  • So, in this exercise, the domain is all real numbers except zero since \(x = 0\) would make the term \(\frac{1}{x}\) undefined.
Vertical Asymptote
A vertical asymptote occurs in the graph of a function when the output values (y-values) increase or decrease without bound as the input value (x-value) approaches a specific point.

In our function \(g(x) = x + \frac{1}{x}\), there is a specific x-value, namely zero, where the function becomes undefined due to division by zero.
  • As \(x\) approaches zero from the positive side, \(g(x)\) increases toward infinity, creating an upward slope.
  • Conversely, as \(x\) approaches zero from the negative side, \(g(x)\) decreases toward negative infinity, which forms a downward slope.

This describes an essential feature known as a vertical asymptote at \(x = 0\). The graph approaches this line but never touches or crosses it, indicating that at \(x = 0\), \(g(x)\) is not defined.
Fractional Functions
Fractional functions involve terms that incorporate fractions, where variables appear in the denominator. These types of functions require close attention, especially to ensure that the denominator does not inadvertently become zero. An undefined case in mathematics occurs when you try to divide by zero, so such values create discontinuities in the function.
  • Consider our function \(g(x) = x + \frac{1}{x}\). The main fractional component, \(\frac{1}{x}\), clearly becomes undefined at \(x = 0\).
  • This means that while evaluating or graphing fractional functions, you must exclude values that make the denominator zero from the domain.

Fractional functions often display unique properties such as vertical asymptotes around the values that make the fraction undefined, and may include horizontal asymptotes as well. Being mindful of these nuances helps in understanding the behavior of the function and its graph.

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Most popular questions from this chapter

A string 28 inches long is to be cut into two pieces. one piece to form a square and the other to form a circle. Express the total area enclosed by the square and circle as a function of the perimeter of the square.

Given that \(f\) is defined for all real numbers, show that the function \(h(x)=f(x)-f(-x)\) is an odd function.

Use a graphing utility to draw several views of the graph of the function. Select the one that most accurately shows the important features of the graph. Give the domain and range of the function. $$f(x)=\left|x^{3}-3 x^{2}-24 x+4\right|$$

Form the composition \(f \circ g\) and give the domain. $$f(x)=x^{2}, \quad g(x)=2 x+5$$

Find the number \((s) x\) in the interval \([0,2 \pi j]\) which satisfy the equation. $$\sin x=-\frac{1}{2}$$
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