Chapter 7.1, Problem 103E

To determine

To prove: that the equation $f\left(x\right)=g\left(x\right)$ is an identity.

Explanation of Solution

Given information:

Two functions, $f$ and $g$ , on a graphing device and their graphs appear identical in the viewing rectangle.

Calculation:

Need to explain the equation $f\left(x\right)=g\left(x\right)$ is an identity, where the graphs of the function $f\left(x\right)$ and $g\left(x\right)$ are identical in the viewing rectangle.

If the graphs of the function $f\left(x\right)$ and $g\left(x\right)$ are identical in the viewing rectangle, the left hand side and the right hand side of the equation $f\left(x\right)$ and $g\left(x\right)$ are equal. Thus, the equation $f\left(x\right)$ and $g\left(x\right)$ is an identity.

To explain this, take an example ${\sin }^{2}x+{\cos }^{2}x=1$ . To prove that this equation is an identity by graphically, the function $f\left(x\right)={\sin }^{2}x+{\cos }^{2}x$ and $g\left(x\right)$ by using a graphing calculator.

To graph $f\left(x\right)={\sin }^{2}x+{\cos }^{2}x$ , enter the equation in the graphing calculator that displays the graph of this function as shown below.

  

To graph $g\left(x\right)=1$ , enter this equation in the graphing calculator that displays the graph of this function as shown below.

  

From the both graph, see that both the functions $f\left(x\right)={\sin }^{2}x+{\cos }^{2}x$ and $g\left(x\right)=1$ have the same graphs in the viewing rectangle of the graphing calculator.

Thus, the equation $f\left(x\right)$ and $g\left(x\right)$ is an identity if the graphs of the function $f\left(x\right)$ and $g\left(x\right)$ are identical in the viewing rectangle.