Chapter 7.1, Problem 18PPS

To determine

The value of $f\circ g\text{ and }g\circ f$ if they exist for the pair of functions $f=\left\{\left(-7,0\right),\left(4,5\right),\left(8,12\right),\left(-3,6\right)\right\}$ and $g=\left\{\left(6,8\right),\left(-12,-5\right),\left(0,5\right),\left(5,1\right)\right\}$ .

Answer to Problem 18PPS

The value of $f\circ g=\left\{\left(6,12\right)\right\}\text{and }g\circ f=\left\{\left(-7,5\right),\left(4,1\right),\left(-3,8\right)\right\}$ .

Explanation of Solution

Given information: The given functions are,

  $\begin{array}{c}f=\left\{\left(-7,0\right),\left(4,5\right),\left(8,12\right),\left(-3,6\right)\right\}\\ g=\left\{\left(6,8\right),\left(-12,-5\right),\left(0,5\right),\left(5,1\right)\right\}\end{array}$

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,

  $\left(\begin{array}{c}g\left(6\right)=8\\ g\left(-12\right)=-5\\ g\left(0\right)=5\\ g\left(5\right)=1\end{array}\right)$

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

  $\left(\begin{array}{c}f\left[g\left(6\right)\right]=f\left(8\right)\\ =12\\ f\left[g\left(-12\right)\right]=f\left(-5\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(0\right)\right]=f\left(5\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(5\right)\right]=f\left(1\right)\mathrm{....}\left(\text{not defined}\right)\end{array}\right)$

Since $-5,5\text{ and }1$ do not lie in domain of $f$ , therefore $f\circ g$ is undefined for the points $-5,5\text{ and }1$ , but $f\left[g\left(-4\right)\right]=4$ . Therefore $f\circ g=\left\{\left(6,12\right)\right\}$ .

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,

  $\left(\begin{array}{c}f\left(-7\right)=0\\ f\left(4\right)=5\\ f\left(8\right)=12\\ f\left(-3\right)=6\end{array}\right)$

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

  $\left(\begin{array}{c}g\left[f\left(-7\right)\right]=g\left(0\right)\\ =5\\ g\left[f\left(4\right)\right]=g\left(5\right)\\ =1\\ g\left[f\left(8\right)\right]=g\left(12\right)\mathrm{....}\left(\text{not defined}\right)\\ g\left[f\left(-3\right)\right]=g\left(6\right)\\ =8\end{array}\right)$

Since 12does not lie in domain of $g$ therefore $g\circ f$ is undefined for the point 12. But $g\left[f\left(-7\right)\right]=5,g\left[f\left(-3\right)\right]=8\text{ and }g\left[f\left(4\right)\right]=1$

Therefore $g\circ f=\left\{\left(-7,5\right),\left(4,1\right),\left(-3,8\right)\right\}$ .