Chapter 7, Problem 26RE

(a)

To determine

Tograph:The functions $f$ and $g$ if $f\left(x\right)=\sin x+\cos x$ and $g\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x}$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\sin x+\cos x$ and $g\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x}$ .

Graph:

Plot the graph of the function $f\left(x\right)=\sin x+\cos x$ as follows:

  

Figure (1)

Plot the graph of the function $g\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x}$ as follows:

  

Figure (2)

Interpretation:From both the graphs it can be interpret that both graphs cuts at same point $\left(0,1\right)$ .

(b)

To determine

To check:The equation $f\left(x\right)=g\left(x\right)$ is an identity or not.

Answer to Problem 26RE

The graphs suggest that $f\left(x\right)=g\left(x\right)$ is not an identity.

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\sin x+\cos x$ and $g\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x}$ .

Calculation:

Graph the functions $f\left(x\right)=\sin x+\cos x$ and $g\left(x\right)=\sqrt{{\sin }^{2}x+{\cos }^{2}x}$ on the same rectangle as shown below.

  

Figure (3)

As shown in the above graph, the graph of both functions is not completely overlapped.It means not every point of the function issame as the corresponding point in another function. So, both the functions are not effectively equal.

Therefore, the graphs suggest that $f\left(x\right)=g\left(x\right)$ is not an identity.