Chapter 7.1, Problem 3CYU

To determine

The value of $f\circ g\text{ and }g\circ f$ if they exist for the pair of functions $f=\left\{\left(2,5\right),\left(6,10\right),\left(12,9\right),\left(7,6\right)\right\}$ and $g=\left\{\left(9,11\right),\left(6,15\right),\left(10,13\right),\left(5,8\right)\right\}$ .

Answer to Problem 3CYU

The value of $f\circ g=\varphi \text{and }g\circ f=\left\{\left(2,8\right),\left(6,13\right),\left(12,11\right),\left(7,15\right)\right\}$ .

Explanation of Solution

Given information: The given functions are,

  $\begin{array}{c}f=\left\{\left(2,5\right),\left(6,10\right),\left(12,9\right),\left(7,6\right)\right\}\\ g=\left\{\left(9,11\right),\left(6,15\right),\left(10,13\right),\left(5,8\right)\right\}\end{array}$

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,

  $\left(\begin{array}{c}g\left(9\right)=11\\ g\left(6\right)=15\\ g\left(10\right)=13\\ g\left(5\right)=8\end{array}\right)$

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

  $\left(\begin{array}{c}f\left[g\left(9\right)\right]=f\left(11\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(6\right)\right]=f\left(15\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(10\right)\right]=f\left(13\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(5\right)\right]=f\left(8\right)\mathrm{....}\left(\text{not defined}\right)\end{array}\right)$

Since $11,15,13\text{ and }8$ do not lie in domain of $f$ , therefore $f\circ g$ is undefined for the points $11,15,13\text{ and }8$ .

Therefore $f\circ g=\varphi$ .

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,

  $\left(\begin{array}{c}f\left(2\right)=5\\ f\left(6\right)=10\\ f\left(12\right)=9\\ f\left(7\right)=6\end{array}\right)$

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

  $\left(\begin{array}{c}g\left[f\left(2\right)\right]=g\left(5\right)\\ =8\\ g\left[f\left(6\right)\right]=g\left(10\right)\\ =13\\ g\left[f\left(12\right)\right]=g\left(9\right)\\ =11\\ g\left[f\left(7\right)\right]=g\left(6\right)\\ =15\end{array}\right)$

Since the points $\left(5,10,9\text{ and 6}\right)$ lie in domain of $g$ therefore $g\circ f$ is defined for all these points.

Therefore $g\circ f=\left\{\left(2,8\right),\left(6,13\right),\left(12,11\right),\left(7,15\right)\right\}$ .