Chapter 1.2, Problem 17E

To determine

To calculate:the values for the sum of the functions $f\left(x\right)=2x+3$ and $g\left(x\right)=3x-5$ , at $x=-2,x=-1,x=0,x=1$ and $x=2$ , graph component piece functions and their sum.

Answer to Problem 17E

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

  $\begin{array}{cccc}x& f\left(x\right)& g\left(x\right)& \left(f+g\right)\left(x\right)\\ -2& -1& -11& -12\\ -1& 1& -8& -7\\ 0& 3& -5& -2\\ 1& 5& -2& 3\\ 2& 7& 1& 8\end{array}$

The graph of the functions $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $\left(f+g\right)\left(x\right)=5x-2$ is provided below,

  

Explanation of Solution

Given information:

The functions, $f\left(x\right)=2x+3$ and $g\left(x\right)=3x-5$ .

Formula used:

The addition of the functions $f$ and $g$ is represented as $f+g$ , mathematically it is denoted by,

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

Graphically the height of addition of two functions is the addition of individual height of the function that is height of first function added to height of second function.

Calculation:

Consider the provided functions, $f\left(x\right)=2x+3$ and $g\left(x\right)=3x-5$ .

Recall the addition of the functions $f$ and $g$ is represented as $f+g$ , mathematically it is denoted by,

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

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

  $f\left(x\right)+g\left(x\right)=2x+3+3x-5$

Group the like terms,

  $\begin{array}{c}f\left(x\right)+g\left(x\right)=2x+3+3x-5\\ =2x+3x+3-5\end{array}$

Combine the like terms,

  $\begin{array}{c}f\left(x\right)+g\left(x\right)=2x+3+3x-5\\ =2x+3x+3-5\\ =5x-2\end{array}$

Therefore, the value of sum of the function is $5x-2$ .

Now, evaluate the value of the individual function $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and sum of the function, $f\left(x\right)+g\left(x\right)=5x-2$ 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, $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $f\left(x\right)+g\left(x\right)=5x-2$ .

  $\begin{array}{c}f\left(-2\right)=2\left(-2\right)+3\\ =-4+3\\ =-1\end{array}$

Next,

  $\begin{array}{c}g\left(-2\right)=3\left(-2\right)-5\\ =-6-5\\ =-11\end{array}$

Last,

  $\begin{array}{c}f\left(-2\right)+g\left(-2\right)=5\left(-2\right)-2\\ =-10-2\\ =-12\end{array}$

For $x=-1$ , substitute the value of x in the functions, $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $f\left(x\right)+g\left(x\right)=5x-2$ .

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

Next,

  $\begin{array}{c}g\left(-1\right)=3\left(-1\right)-5\\ =-3-5\\ =-8\end{array}$

Last,

  $\begin{array}{c}f\left(-1\right)+g\left(-1\right)=5\left(-1\right)-2\\ =-5-2\\ =-7\end{array}$

For $x=0$ , substitute the value of x in the functions, $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $f\left(x\right)+g\left(x\right)=5x-2$ .

  $\begin{array}{c}f\left(0\right)=2\left(0\right)+3\\ =3\end{array}$

Next,

  $\begin{array}{c}g\left(0\right)=3\left(0\right)-5\\ =-5\end{array}$

Last,

  $\begin{array}{c}f\left(0\right)+g\left(0\right)=5\left(0\right)-2\\ =-2\end{array}$

For $x=1$ , substitute the value of x in the functions, $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $f\left(x\right)+g\left(x\right)=5x-2$ .

  $\begin{array}{c}f\left(1\right)=2\left(1\right)+3\\ =2+3\\ =5\end{array}$

Next,

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

Last,

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

For $x=2$ , substitute the value of x in the functions, $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $f\left(x\right)+g\left(x\right)=5x-2$ .

  $\begin{array}{c}f\left(2\right)=2\left(2\right)+3\\ =4+3\\ =7\end{array}$

Next,

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

Last,

  $\begin{array}{c}f\left(2\right)+g\left(2\right)=5\left(2\right)-2\\ =10-2\\ =8\end{array}$

Thus, the above results are tabulated as,

  $\begin{array}{cccc}x& f\left(x\right)& g\left(x\right)& \left(f+g\right)\left(x\right)\\ -2& -1& -11& -12\\ -1& 1& -8& -7\\ 0& 3& -5& -2\\ 1& 5& -2& 3\\ 2& 7& 1& 8\end{array}$

Plot the graph of the functions $f\left(x\right)=2x+3$ , $g\left(x\right)=3x-5$ and $\left(f+g\right)\left(x\right)=5x-2$ on the same coordinate axes and visualize the each of them.

  

Thus, the results obtained graphically and mathematically are same.