Mastery Check Solutions

1. Marisol and Nicole were asked to compose h with f.  Name the person that has the composition of functions correctly written, then compose the function with the domain restrictions.
   
   

   Marisol	Nicole
   $\left[f\circ h\right]\left(x\right)=f\left[h\left(x\right)\right]$	$\left[h\circ f\right]\left(x\right)=h\left[f\left(x\right)\right]$

Nicole
$\begin{array}{rcl}\left[h\circ f\right]\left(x\right)& =& h\left[f\left(x\right)\right] \\ & =& \frac{1}{\sqrt{x}–6}\left(\frac{\sqrt{x}+6}{\sqrt{x}+6}\right) {\mathrm{Domain}}_f: \left\{x|x\in \mathrm{ℝ}, x\ge 0\right\} \\ & =& \frac{\sqrt{x}+6}{x–36}, x\ne 36, x\ge 0\end{array}$

2. Marisol and Nicole believe that g and h are inverses. Show all work to determine if they are correct.

$\begin{array}{ccc}\left[g\circ h\right]\left(x\right)=g\left[h\left(x\right)\right]& & \left[h\circ g\right]\left(x\right)=g\left[h\left(x\right)\right]\\ =\frac{1}{\left(\frac{1}{x–6}\right)}+6& & =\frac{1}{\left(\frac{1}{x}+6\right)–6}\\ =\left(1÷\frac{1}{x–6}\right)+6& & =\frac{1}{\frac{1}{x}}=1÷\frac{1}{x}\\ =x–6+6& & =x\\ =x& & \end{array}$

Marisol and Nicole are correct since the result of both compositions is x.

3. Marisol and Nicole’s instructor told them$j\left(x\right)= x–6$ and is composed with a new function k and one of the given functions f, g, or h.
   
   Decompose j(x) to determine k(x) and one of the given functions. Show your work to prove you are correct.

$\begin{array}{rcl}k\left(x\right)& =& \frac{1}{x} h\left(x\right)=\frac{1}{x–6}\\ j\left(x\right)& =& \left[k\circ h\right]\left(x\right)\\ & =& k\left[h\left(x\right)\right]=\frac{1}{\frac{1}{x–6}}\\ & =& 1÷\left(\frac{1}{x–6}\right)=1·\left(x–6\right)\\ j\left(x\right)& =& x–6\end{array}$