Question

a. Plot the graph of \(f(x)=x / x\) and \(g(x)=1\). b. Are the functions \(f\) and \(g\) identical? Why or why not?

Short answer

The graphs of the functions \(f(x)=\frac{x}{x}\) and \(g(x)=1\) seem almost the same, but they are not identical. The function \(f(x)\) is undefined at \(x=0\), while the function \(g(x)\) includes this point.

Step-by-step solution

Analyze the functions

Before plotting, we need to understand the functions better. The given functions are: 1. \(f(x)=\frac{x}{x}\) 2. \(g(x)=1\)

Simplify the functions

We can simplify the function \(f(x)\) as follows: \(f(x)=\frac{x}{x}\) Since the numerator and denominator are the same and nonzero, we can simplify it to: \(f(x)=1\) However, we must take into consideration that the function is undefined for \(x=0\), as we cannot have 0 in the denominator. So, \(f(x)=1\) for all \(x \neq 0\).

Plot the graphs

Now, we will plot the graphs of both functions: 1. For \(f(x)=\frac{x}{x}\): The function has a value of 1 for all \(x \neq 0\). At \(x=0\), the function is undefined. We can plot a continuous straight line at \(y=1\) for all values of \(x\) except the point (0, 1). 2. For \(g(x)=1\): The function is simply a straight line at \(y=1\) for all values of \(x\).

Compare the functions and determine if they are identical

Based on the graph, the two functions appear to be the same, except for one small difference: 1. The function \(f(x)=\frac{x}{x}\) is equal to 1 for all \(x \neq 0\) and undefined at \(x=0\). 2. The function \(g(x)=1\) is equal to 1 for all \(x\). Thus, while they appear the same on the graph, the functions are not identical due to the undefined point at \(x=0\) in the function \(f(x)\). In conclusion, although the graphs of the functions \(f(x)=\frac{x}{x}\) and \(g(x)=1\) seem almost the same, the two functions are not identical. The function \(f(x)\) is undefined at \(x=0\), while the function \(g(x)\) includes this point.