Chapter 6.1, Problem 17PPS

To determine

The value of $f\circ g\text{ and }g\circ f$ if they exist for the pair of functions $f=\left\{\left(-8,-4\right),\left(0,4\right),\left(2,6\right),\left(-6,-2\right)\right\}$ and $g=\left\{\left(4,-4\right),\left(-2,-1\right),\left(-4,0\right),\left(6,-5\right)\right\}$ .

Answer to Problem 17PPS

The value of $f\circ g=\left\{\left(-4,4\right)\right\}\text{and }g\circ f=\left\{\left(-8,0\right),\left(0,-4\right),\left(2,-5\right),\left(-6,-1\right)\right\}$ .

Explanation of Solution

Given information: The given functions are,

  $\begin{array}{c}f=\left\{\left(-8,-4\right),\left(0,4\right),\left(2,6\right),\left(-6,-2\right)\right\}\\ g=\left\{\left(4,-4\right),\left(-2,-1\right),\left(-4,0\right),\left(6,-5\right)\right\}\end{array}$

Calculation: To evaluate $f\circ g$ first evaluate $g\left(x\right)$ and then use range of $g\left(x\right)$ as domain of $f\left(x\right)$ to evaluate $f\circ g$ . Therefore $g\left(x\right)$ is evaluated as,

  $\left(\begin{array}{c}g\left(4\right)=-4\\ g\left(-2\right)=-1\\ g\left(-4\right)=0\\ g\left(6\right)=-5\end{array}\right)$

Now, evaluating $f\circ g$ using the range of $g\left(x\right)$ as domain of $f\left(x\right)$ .

  $\begin{array}{c}f\left[g\left(4\right)\right]=f\left(-4\right)\text{ which is undefined}\\ f\left[g\left(-2\right)\right]=f\left(-1\right)\text{ also undefined}\\ f\left[g\left(-4\right)\right]=f\left(0\right)\\ =4\\ f\left[g\left(6\right)\right]=f\left(-5\right)\text{also undefined}\end{array}$

Since $-4,-1\text{ and}-5$ do not lie in domain of $f$ , therefore $f\circ g$ is undefined for the points $-4,-1\text{ and}-5$ , but $f\left[g\left(-4\right)\right]=4$ . Therefore $f\circ g=\left\{\left(-4,4\right)\right\}$ .

Similarly to find $g\circ f$ evaluate $f\left(x\right)$ first and then use range of $f\left(x\right)$ as domain of $g\left(x\right)$ to evaluate $g\circ f$ . Therefore $f\left(x\right)$ is evaluated as,

  $\left(\begin{array}{c}f\left(-8\right)=-4\\ f\left(0\right)=4\\ f\left(2\right)=6\\ f\left(-6\right)=-2\end{array}\right)$

Now, evaluating $g\circ f$ using the range of $f\left(x\right)$ as domain of $g\left(x\right)$ .

  $\begin{array}{c}g\left[f\left(-8\right)\right]=g\left(-4\right)\\ =0\\ g\left[f\left(0\right)\right]=g\left(4\right)\\ =-4\\ g\left[f\left(2\right)\right]=g\left(6\right)\\ =-5\\ g\left[f\left(-6\right)\right]=g\left(-2\right)\\ =-1\end{array}$

Since $g\left[f\left(-8\right)\right]=0,g\left[f\left(0\right)\right]=-4,g\left[f\left(2\right)\right]=-5\text{ and }g\left[f\left(-6\right)\right]=-1$ .

Therefore $g\circ f=\left\{\left(-8,0\right),\left(0,-4\right),\left(2,-5\right),\left(-6,-1\right)\right\}$ .