Question

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{x^{3}}{x^{3}+1} $$

Short answer

An appropriate viewing window is [-5, 5] on the x-axis and [-1, 2] on the y-axis. The graph shows symmetry about the origin, intercepts at (0, 0), an asymptote at \(y = 1\), and approaches the asymptote for both positive and negative large x-values. The domain of the function is \((-∞, ∞)\) and the range is \((-∞, 1)\).

Step-by-step solution

Find the domain of the function

The function to be graphed is \(f(x) = \frac{x^3}{x^3 + 1}\). The function is defined everywhere since the denominator will never be equal to zero. Therefore, the domain of the function is \(\text{Domain}(f) = (-\infty, \infty)\).

Determine the range of the function

In order to find the range, we need to look at the function's limit properties. As x approaches negative infinity, the limit of the function is 1, and as x approaches positive infinity, the limit is 1 as well: \(\lim_{x\to -\infty} \frac{x^3}{x^3+1} = \lim_{x\to -\infty} \frac{1}{\frac{1}{x^{3}}+1} = 1\) \(\lim_{x\to \infty} \frac{x^3}{x^3+1} = \lim_{x\to \infty} \frac{1}{\frac{1}{x^{3}}+1} = 1\) Therefore, the function will never reach the value 1. So, the range of the function is \(\text{Range}(f) = (-\infty, 1)\).

Calculate the x and y-intercepts

To find the x-intercept, we set f(x) equal to zero: \(0 = \frac{x^3}{x^3 + 1}\) By observing that the numerator must be equal to zero, we find the x-intercept: \(x^3 = 0 \Rightarrow x = 0\) Thus, the x-intercept is a single point at (0, 0). To find the y-intercept, we set x equal to 0: \(f(0) = \frac{0^3}{0^3 + 1} = 0\) So the y-intercept is also at the origin (0, 0).

Observe the properties of the function

By analyzing the behavior of the function, we notice the function is odd, which means it is symmetric about the origin: \(f(-x) = \frac{(-x)^3}{(-x)^3+1} = -\frac{x^3}{x^3+(-1)} = -f(x)\)

Choose an appropriate viewing window and plot the graph

Now with the information we've gathered, we can choose an appropriate viewing window and plot the graph. Let's choose a window of [-5, 5] on the x-axis, and [-1, 2] on the y-axis. This window shows the function's behavior within a reasonable range for both the x and y-values. The graph will show symmetry about the origin, intercepts at (0, 0), an asymptote at y = 1, and it will approach the asymptote for both positive and negative large x-values.