Chapter 2.8 Practice

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Fully Worked Out Questions

1. Take \(f\) to be the function that is given by \[f(x) = 2x + 5.\] Find equations for the functions given by \(R(f)\), \(M_y(f)\), and \(M_x(f)\) and sketch \(f\) along with these transformed functions.

   Answer

   Calculate to get \[R(f)=-\left(2(-x)+5\right)=2x-5,\] \[M_y(f)=2(-x)+5=-2x+5\] \[M_x(f)=-\left(2x+5\right)=-2x-5.\] The sketches of \(f\), \(R(f)\), \(M_y(f)\) and \(M_x(f)\) are shown below:
   
   
   
   

2. Sketch the function \(f\), where \[f(x) = \frac{1}{(x+3)}.\] Hint: think of transformations, compare \(f\) to \(g(x)=\frac{1}{x}\).

   Answer

   The function \(f\) can be rewritten like this: \[f=g\circ T_{3},\] so the sketch of \(f\) looks like this

3. Sketch the function \(f\), where \[f(x) = \frac{1}{(x-4)^3}.\] Hint: think of transformations, compare \(f\) to \(g(x)=\frac{1}{x^3}\).

   Answer

   The function \(f\) can be rewritten like this: \[f=g\circ T_{-4},\] so the sketch of \(f\) looks like this

4. Sketch the function \(f\), where \[f(x) = \frac{1}{(x-3)^4}.\] Hint: think of transformations, compare \(f\) to \(g(x)=\frac{1}{x^4}\).

   Answer

   The function \(f\) can be rewritten like this: \[f=g\circ T_{3},\] so the sketch of \(f\) looks like this

5. Define \(\mathrm{Recip}(x)=\frac{1}{x}.\) Use \(y\)-axis inversion to sketch \(\mathrm{Recip}\circ(f)\) where \(f\) is this function:

Answer

The sketch of \(\mathrm{Recip}\circ(f)\) looks like this

Questions

1. For each of these functions \(f\), sketch \(f\), determine a formula for \(R(f), M_y(f), M_x(f)\) and sketch these as well.
   1. \(f(x)=-2x+1\)
   2. \(f(x)=x^2+1\)
   3. \(f(x)=|x+2|-1\)
   4. \(f(x)=\sqrt{x+1}\)
2. Define \(\mathrm{Recip}(x)=\frac{1}{x}.\) Use \(y\)-axis inversion to sketch \(\mathrm{Recip}\circ(f)\) where \(f\) is this function:

Answers

1. 1. \[R(f)=-2x-1,\] \[M_y(f)=2x+1\] \[M_x(f)=2x-1.\] The sketches of \(f\), \(R(f)\), \(M_y(f)\) and \(M_x(f)\) are shown below:
      
      
      
      
   2. \[R(f)=-x^2-1,\] \[M_y(f)1=x^2+1\] \[M_x(f)=-x^2-1.\] The sketches of \(f\), \(R(f)\), \(M_y(f)\) and \(M_x(f)\) are shown below:
      
      
      
      
   3. \[R(f)=-|-x+2|+1,\] \[M_y(f)=|-x+2|-1\] \[M_x(f)=-|x+2|+1.\] The sketches of \(f\), \(R(f)\), \(M_y(f)\) and \(M_x(f)\) are shown below:
      
      
      
      
   4. \[R(f)=-\sqrt{-x+1},\] \[M_y(f)=\sqrt{-x+1}\] \[M_x(f)=-\sqrt{x+1}.\] The sketches of \(f\), \(R(f)\), \(M_y(f)\) and \(M_x(f)\) are shown below:
      
      
      
      

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