Chapter 1.2, Problem 23E

To determine

To calculate: For the product of the functions $F\left(x\right)=x^2+1$ and $G\left(x\right)=x+1$ , the values at $x=-2,x=-1,x=0,x=1$ and $x=2$ , graph component piece functions and their product.

Answer to Problem 23E

The value of the functions and product of the functions for different values of x are,

  $\begin{array}{cccc}x& F\left(x\right)& G\left(x\right)& \left(F\cdot G\right)\left(x\right)\\ -2& 5& -1& -5\\ -1& 2& 0& 0\\ 0& 1& 1& 1\\ 1& 2& 2& 4\\ 2& 5& 3& 15\end{array}$

The graph of the functions $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $\left(F\cdot G\right)\left(x\right)=x^3+x^2+x+1$ is provided below,

  

Explanation of Solution

Given information:

The functions, $F\left(x\right)=x^2+1$ and $G\left(x\right)=x+1$ .

Formula used:

The multiplication or product of the functions $f$ and $g$ is represented as $f\cdot g$ , mathematically it is denoted by,

  $\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$

Calculation:

Consider the provided functions, $F\left(x\right)=x^2+1$ and $G\left(x\right)=x+1$ .

Recall the multiplication or product of the functions $f$ and $g$ is represented as $f\cdot g$ , mathematically it is denoted by,

  $\left(f\cdot g\right)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$

Evaluate the value of $f\left(x\right)\cdot g\left(x\right)$ that is product of the functions,

  $F\left(x\right)\cdot G\left(x\right)=\left(x^2+1\right)\cdot \left(x+1\right)$

Multiply the first term of the first bracket with the terms in second bracket and do the same with the second term of the first bracket.

  $\begin{array}{c}F\left(x\right)\cdot G\left(x\right)=\left(x^2+1\right)\cdot \left(x+1\right)\\ =x^2\left(x+1\right)+1\left(x+1\right)\\ =x^3+x^2+x+1\end{array}$

Therefore, the value of product of the function is $x^3+x^2+x+1$

  $x^2+x+2$ .

Now, evaluate the value of the individual function $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and product of the functions, $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ at the provided points $x=-2,x=-1,x=0,x=1$ and $x=2$ .

For $x=-2$ , substitute the value of x in the functions, $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ .

  $\begin{array}{c}F\left(-2\right)={\left(-2\right)}^2+1\\ =4+1\\ =5\end{array}$

Next,

  $\begin{array}{c}G\left(-2\right)=-2+1\\ =-1\end{array}$

Last,

  $\begin{array}{c}F\left(-2\right)\cdot G\left(-2\right)={\left(-2\right)}^3+{\left(-2\right)}^2+\left(-2\right)+1\\ =-8+4-2+1\\ =-5\end{array}$

For $x=-1$ , substitute the value of x in the functions, $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ .

  $\begin{array}{c}F\left(-1\right)={\left(-1\right)}^2+1\\ =1+1\\ =2\end{array}$

Next,

  $\begin{array}{c}G\left(-1\right)=-1+1\\ =0\end{array}$

Last,

  $\begin{array}{c}F\left(-1\right)\cdot G\left(-1\right)={\left(-1\right)}^3+{\left(-1\right)}^2+\left(-1\right)+1\\ =-1+1-1+1\\ =0\end{array}$

For $x=0$ , substitute the value of x in the functions, $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ .

  $\begin{array}{c}F\left(0\right)={\left(0\right)}^2+1\\ =1\end{array}$

Next,

  $\begin{array}{c}G\left(0\right)=0+1\\ =1\end{array}$

Last,

  $\begin{array}{c}F\left(0\right)\cdot G\left(0\right)={\left(0\right)}^3+{\left(0\right)}^2+\left(0\right)+1\\ =1\end{array}$

For $x=1$ , substitute the value of x in the functions, $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ .

  $\begin{array}{c}F\left(1\right)={\left(1\right)}^2+1\\ =2\end{array}$

Next,

  $\begin{array}{c}G\left(1\right)=1+1\\ =2\end{array}$

Last,

  $\begin{array}{c}F\left(1\right)\cdot G\left(1\right)={\left(1\right)}^3+{\left(1\right)}^2+\left(1\right)+1\\ =1+1+1+1\\ =4\end{array}$

For $x=2$ , substitute the value of x in the functions, $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $F\left(x\right)\cdot G\left(x\right)=x^3+x^2+x+1$ .

  $\begin{array}{c}F\left(2\right)={\left(2\right)}^2+1\\ =4+1\\ =5\end{array}$

Next,

  $\begin{array}{c}G\left(2\right)=2+1\\ =3\end{array}$

Last,

  $\begin{array}{c}F\left(2\right)\cdot G\left(2\right)={\left(2\right)}^3+{\left(2\right)}^2+\left(2\right)+1\\ =8+4+2+1\\ =15\end{array}$

Thus, the above results are tabulated as,

  $\begin{array}{cccc}x& F\left(x\right)& G\left(x\right)& \left(F\cdot G\right)\left(x\right)\\ -2& 5& -1& -5\\ -1& 2& 0& 0\\ 0& 1& 1& 1\\ 1& 2& 2& 4\\ 2& 5& 3& 15\end{array}$

Plot the graph of the functions $F\left(x\right)=x^2+1$ , $G\left(x\right)=x+1$ and $\left(F\cdot G\right)\left(x\right)=x^3+x^2+x+1$ on the same coordinate axes and visualize the each of them.

  

Thus, the results obtained graphically and mathematically are same.