Chapter 1.2, Problem 69E

(a)

To determine

To find the value of given financial composition function and to plot the graph using the obtained values

Answer to Problem 69E

The graph of the function is drawn based on data.

Explanation of Solution

Given information:

  $\left(f?g\right)(\times )and\left(g?f\right)(\times )iff(\times )=\sin (\times )andg(\times )=x^2$

Calculation:

Given that the functions

  $\begin{array}{l}f\left(x\right)=\sin \left(x\right)\\ g\left(x\right)=x^2\end{array}$

Here, we are to find the functional compositions $\left(f\circ g\right)$ and $\left(g\circ f\right)\left(x\right)$ .

The composition $f\circ g$ of functions $f$ and $g$ is a function defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ and the composition go $g\circ f$ functions $f$ and $g$ is a function defined by $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Therefore, the functional composition $\left(f\circ g\right)\left(x\right)$ is

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\left[Thedefinition\right]\\ =f\left(x^2\right)\left[WriteouttheformulafortheinnerfunctionG\right]\\ =\sin \left(x^2\right)\left[PlugtheformulaforG\text{into}\text{the outer function F}\right]\end{array}$

Therefore, the functional composition $\left(g\circ f\right)\left(x\right)$ is

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\left[Thedefinition\right]\\ =g\left(\sin x\right)\left[WriteouttheformulafortheouterfunctionF\right]\\ ={\left(\sin x\right)}^2\left[PlugtheformulaforF\text{into}\text{the inner function G}\right]\\ ={\sin }^{2}x\left[Expandthesquare\right]\end{array}$

The following functions are plotted below:

  

(b)

To determine

To find the value of given financial composition function and to plot the graph using the obtained values

Answer to Problem 69E

The graph of the function is drawn based on data.

Explanation of Solution

Given information:

  $\left(f?g\right)(\times )and\left(g?f\right)(\times )iff(\times )=exandg(\times )=x^2$

Calculation:

Given that the functions

  $\begin{array}{l}f\left(x\right)=e^x\\ g\left(x\right)=x^2\end{array}$

Here, we are to find the functional compositions $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ .

The composition $f\circ g$ of functions $f$ and $g$ is a function defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ and the composition go $g\circ f$ functions $f$ and $g$ is a function defined by $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Therefore, the functional composition $\left(f\circ g\right)\left(x\right)$ is

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\left[Thedefinition\right]\\ =f\left(x^2\right)\left[WriteouttheformulafortheinnerfunctionG\right]\\ =e^{x^2}\left[PlugtheformulaforG\text{into}\text{the outer function F}\right]\end{array}$

Therefore, the functional composition $\left(g\circ f\right)\left(x\right)$ is

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\left[Thedefinition\right]\\ =g\left(e^x\right)\left[WriteouttheformulafortheouterfunctionF\right]\\ ={\left(e^x\right)}^2\left[PlugtheformulaforF\text{into}\text{the inner function G}\right]\\ =e^{2x}\left[Expandthesquare\right]\end{array}$

The following functions are plotted below:

  

(c)

To determine

To find the value of given financial composition function and to plot the graph using the obtained values

Answer to Problem 69E

The graph of the function is drawn based on data.

Explanation of Solution

Given information:

  $\left(f?g\right)(\times )and\left(g?f\right)(\times )iff(\times )=exandg(\times )=\sin \left(x\right)$

Calculation:

Given that the functions

  $\begin{array}{l}f\left(x\right)=e^x\\ g\left(x\right)=\sin \left(x\right)\end{array}$

Here, we are to find the functional compositions $\left(f\circ g\right)\left(x\right)$ and $\left(g\circ f\right)\left(x\right)$ .

The composition $f\circ g$ of functions $f$ and $g$ is a function defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ and the composition go $g\circ f$ functions $f$ and $g$ is a function defined by $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Therefore, the functional composition $\left(f\circ g\right)\left(x\right)$ is

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)\left[Thedefinition\right]\\ =f\left(\sin \left(x\right)\right)\left[WriteouttheformulafortheinnerfunctionG\right]\\ =e^{\sin \left(x\right)}\left[PlugtheformulaforG\text{into}\text{the outer function F}\right]\end{array}$

Therefore, the functional composition $\left(g\circ f\right)\left(x\right)$ is

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\left[Thedefinition\right]\\ =g\left(e^x\right)\left[WriteouttheformulafortheouterfunctionF\right]\\ =\sin \left(e^x\right)\left[PlugtheformulaforF\text{into}\text{the inner function G}\right]\\ \end{array}$

The following functions are plotted below: