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Chapter 2: Problem 46

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$g(x)=\frac{1}{x-2}$$

Short Answer

Expert verified
The graph of the function \(g(x) = \frac{1}{x-2}\) is a hyperbola that has been shifted 2 units to the right from the graph of the base function \(f(x) = \frac{1}{x}\). This shift introduces a vertical asymptote at \(x = 2\). The graph also has a horizontal asymptote at \(y = 0\).

Step by step solution

01

Analyze the Function

Given the function \(g(x) = \frac{1}{x-2}\), it can be observed that this function is a transformation of the base function \(f(x) = \frac{1}{x}\). In this case, it's a horizontal shift of 2 units to the right.
02

Apply the Horizontal Shift

A horizontal shift in the function can be noted when we compare the input value (x) of the function with the base function. If there is a subtraction or addition in the input (x) of the function, it implies there is a horizontal shift. In our function \(g(x) = \frac{1}{x-2}\), there's a subtraction of 2 from x, thus there's a horizontal shift to the right by 2 units. This means every point on the base function \(f(x) = \frac{1}{x}\) would move 2 units to the right.
03

Graph the Function

Begin by plotting the base function \(f(x) = \frac{1}{x}\), which is a hyperbola centered at the origin. Afterward, shift all points 2 units to the right to graph the final function \(g(x) = \frac{1}{x-2}\). The resultant graph will still be a hyperbola, but this time shifted to the right by 2 units, introducing a vertical asymptote at \(x = 2\).
04

Identify Key Components

Looking at the final graph of \(g(x) = \frac{1}{x-2}\), one can identify key components such as the vertical asymptote at \(x = 2\) and the horizontal asymptote at \(y = 0\). The function is undefined at \(x = 2\), hence it represents the vertical asymptote. Knowing these features will help understand the behavior of the function and express its key characteristics correctly.

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Most popular questions from this chapter

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=3,\) a horizontal asymptote \(y=0, y\) -intercept at \(-1,\) and no \(x\) -intercept.

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\)-intercept.

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\).
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