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Chapter 2: Problem 48

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

Short Answer

Expert verified
To graph \(h(x) = \frac{1}{x}+1\), start by graphing the base function \(f(x) = \frac{1}{x}\). Next, vertically shift every point on this graph up by one unit to represent the '+1' in the function \(h(x)\).

Step by step solution

01

Identify the Base Function

The base function here is \(f(x) = \frac{1}{x}\). This function has a hyperbolic shape starting in the third quadrant and extending to the first quadrant. The function \(h(x)\) is a transformation of this base function.
02

Identify the Transformation

The function \(h(x) = \frac{1}{x}+1\) is a vertical transformation of the base function \(f(x) = \frac{1}{x}\). The '+1' shifts the base function up by 1 unit. Specifically, for all \(x\) values, the \(y\) value of \(h(x)\) will be one unit higher than the \(y\) value at the same \(x\) for the base function.
03

Graph the Function

To graph \(h(x)\), start by sketching the base function \(f(x) = \frac{1}{x}\). Then, shift every point on this base function up by one unit. The resulting graph represents the function \(h(x) = \frac{1}{x}+1\).

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Most popular questions from this chapter

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{3}+1}{x^{2}+2 x}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{5 x^{2}}{x^{2}-4} \cdot \frac{x^{2}+4 x+4}{10 x^{3}}$$

Will help you prepare for the material covered in the next section. Solve: \(2 x^{2}+x=15\)

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a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+4}{x}$$
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