2-12 Properties of Rational Functions

Identify the domain of the function.
 $f\left(x\right)=\frac{1}{x+4}+3$  

•  $\left(-\infty,\ -4\right)\cup\left(-4,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$  

•  $\left(-\infty,\ -4\right)\cup\left(-4,\ 3\right)\cup\left(3,\ \infty\right)$   

Identify the range of the function.
 $f\left(x\right)=\frac{1}{x+4}+3$  

•  $\left(-\infty,\ -4\right)\cup\left(-4,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$  

•  $\left(-\infty,\ -4\right)\cup\left(-4,\ 3\right)\cup\left(3,\ \infty\right)$   

Identify the range of the function.
 $f\left(x\right)=-\frac{2}{x-3}+1$  

•  $\left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$  

•  $\left(-\infty,\ -2\right)\cup\left(-2,\ \infty\right)$   

Identify the domain of the function.
 $f\left(x\right)=-\frac{2}{x-3}+1$  

•  $\left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$  

•  $\left(-\infty,\ -2\right)\cup\left(-2,\ \infty\right)$   

Describe the transformation of the function from the basic function  $y=\frac{1}{x}$ .
 $f\left(x\right)=-\frac{2}{x-3}+1$  

• Shift 3 units right, vertical stretch by factor 2, reflection over the x-axis, shift 1 unit up

• Shift 3 units right, horizontal stretch by factor 2, reflection over the x-axis, shift 1 unit up

• Shift 1 unit right, vertical stretch by factor 2, reflection over the y-axis, shift 3 units down

• Shift 1 unit left, horizontal stretch by factor 2, reflection over the y-axis, shift 3 units up

Describe the transformation of the function from the basic function  $y=\frac{1}{x}$ .
 $f\left(x\right)=\frac{5x+9}{x+1}$  

• Shift 1 unit left, vertical stretch by factor 4, shift 5 units up

• Shift 1 unit right, vertical stretch by factor 4, shift 5 units up

• Shift 1 unit left, vertical stretch by factor 4, reflection over the x-axis, shift 5 units up

• Shift 5 units right, horizontal stretch by factor 4, shift 1 unit up

Describe the transformation of the function from the basic function  $y=\frac{1}{x}$ .
 $f\left(x\right)=\frac{2x+8}{x+3}$  

• Shift 3 units right, vertical stretch by factor 2, shift 2 units up

• Shift 2 units right, vertical stretch by factor 2, shift 3 units up

• Shift 3 units right, horizontal stretch by factor 2, shift 2 units down

• Shift 3 units left, vertical stretch by factor 2, shift 2 units up

 Identify the domain. 
 $f\left(x\right)=\frac{x^2+5x+6}{2x^2+4x}$  

•  $\left(-\infty,\ -2\right)$  

•  $\left(-\infty,\ -2\right)\cup\left(-2,\ 0\right)\cup\left(0,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ 1\right)\cup\left(1,\ 4\right)\cup\left(4,\ \infty\right)$  

 Identify the domain. 
 $f\left(x\right)=\frac{x^2+4x+3}{x^2-\ x\ -\ 6}$  

•  $\left(-\infty,\ 6\right)$  

•  $\left(-\infty,\ -2\right)\cup\left(-2,\ 3\right)\cup\left(3,\ \infty\right)$  

•  $\left(-\infty,\ \infty\right)$  

•  $\left(-\infty,\ -3\right)\cup\left(-3,\ -1\right)\cup\left(-1,\ \infty\right)$  

 Identify any vertical asymptotes. 
 $f\left(x\right)=\frac{1}{x^2-\ 3x\ }$  

•  $x=0$  
   $x=3$  
  

•  $x=0$  
   $x=-3$  
  

•  $y=0$  
   $y=3$  
  

• There are no vertical asymptotes.