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Chapter 3: Problem 48

In Exercises \(45-56,\) use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$ g(x)=\frac{1}{x+1}-2 $$

Short Answer

Expert verified
The graph of the function \(g(x)=\frac{1}{x+1}-2\) is a hyperbola shifted to the left by 1 unit and shifted downward by 2 units.

Step by step solution

01

Identify base function and transformations

The given function \(g(x) = \frac{1}{x+1}-2\) can be rewritten as \(f(x-(-1))-2\). Comparing this form with \(f(x) = \frac{1}{x}\), the base function is \(f(x) = \frac{1}{x}\). There is a horizontal shift to the left by 1 unit represented by +1 inside the function. And there is a vertical shift downward by 2 units represented by -2 outside the function.
02

Sketch base function

Knowing that the base function is \(f(x) = \frac{1}{x}\), the rough sketch of this function will be a hyperbola. On the xy-plane, \(f(x) = \frac{1}{x}\) approaches but never reaches zero (asymptote). It will be positive in the first and third quadrants, and negative in the second and fourth quadrants.
03

Apply transformations

Based on the identified transformations, shift the graph of the base function \(f(x) = \frac{1}{x}\) left by 1 unit and then downward by 2 units. Note, there is no vertical or horizontal reflections as well as no vertical or horizontal stretches.
04

Final Graph

The final graph of \(g(x)=\frac{1}{x+1}-2\) will be the graph from the previous step. It should be a hyperbola which is shifted to the left 1 unit and downward 2 units.

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