Chapter 7.1, Problem 4CYU

To determine

The value of $f\circ g\text{ and }g\circ f$ if they exist for the pair of functions $f=\left\{\left(-5,4\right),\left(14,8\right),\left(12,1\right),\left(0,-3\right)\right\}$ and $g=\left\{\left(-2,-4\right),\left(-3,2\right),\left(-1,4\right),\left(5,-6\right)\right\}$ .

Answer to Problem 4CYU

The value of $f\circ g=\varphi \text{and }g\circ f=\left\{\left(0,2\right)\right\}$ .

Explanation of Solution

Given information: The given functions are,

  $\begin{array}{c}f=\left\{\left(-5,4\right),\left(14,8\right),\left(12,1\right),\left(0,-3\right)\right\}\\ g=\left\{\left(-2,-4\right),\left(-3,2\right),\left(-1,4\right),\left(5,-6\right)\right\}\end{array}$

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,

  $\left(\begin{array}{c}g\left(-2\right)=-4\\ g\left(-3\right)=2\\ g\left(-1\right)=4\\ g\left(5\right)=-6\end{array}\right)$

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

  $\left(\begin{array}{c}f\left[g\left(-2\right)\right]=f\left(-4\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(-3\right)\right]=f\left(2\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(-1\right)\right]=f\left(4\right)\mathrm{....}\left(\text{not defined}\right)\\ f\left[g\left(5\right)\right]=f\left(-6\right)\mathrm{....}\left(\text{not defined}\right)\end{array}\right)$

Since all these points $\left(-4,2,4\text{ and}-6\right)$ do not lie in domain of $f$ , therefore $f\circ g$ is undefined for the points $\left(-4,2,4\text{ and}-6\right)$ .

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,

  $\left(\begin{array}{c}f\left(-5\right)=4\\ f\left(14\right)=8\\ f\left(12\right)=1\\ f\left(0\right)=-3\end{array}\right)$

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

  $\left(\begin{array}{c}g\left[f\left(-5\right)\right]=g\left(4\right)\mathrm{....}\left(\text{not defined}\right)\\ g\left[f\left(14\right)\right]=g\left(8\right)\mathrm{....}\left(\text{not defined}\right)\\ g\left[f\left(12\right)\right]=g\left(1\right)\mathrm{....}\left(\text{not defined}\right)\\ g\left[f\left(0\right)\right]=g\left(-3\right)\\ =2\end{array}\right)$

Since the points $\left(4,8\text{ and 1}\right)$ do not lie in domain of $g$ therefore $g\circ f$ is defined for all these points $\left(4,8\text{ and 1}\right)$ but $g\left[f\left(0\right)\right]=2$ .

Therefore $g\circ f=\left\{\left(0,2\right)\right\}$ .