Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=\frac{1}{4} f(x)=\frac{2}{x^{3}}$$

Step-by-Step Solution

Answer

The graph of \(g(x)=\frac{2}{x^{3}}\) is simply a vertical shrink of the graph of \(f(x)=\frac{8}{x^{3}}\) by a factor of 1 / 4. This means that \(g(x)\) is smaller than \(f(x)\) for any chosen \(x\) value, however they both exhibit the same general shape and behavior.

Step 1: Evaluate each function at x = 2

Start by plugging \(x = 2\) into each function: \[f(2)=\frac{8}{2^3}=\frac{8}{8}=1,\] \[g(2)=\frac{2}{2^3}=\frac{2}{8}=0.25.\] You can see that \(g(2)\) is \(\frac{1}{4}\) the size of \(f(2)\).

Step 2: Evaluate each function at x = -2

Next, plug \(x = -2\) into each function: \[f(-2)=\frac{8}{-2^3}=\frac{8}{-8}=-1,\] \[g(-2)=\frac{2}{-2^3}=\frac{2}{-8}=-0.25.\] Again, \(g(-2)\) is \(\frac{1}{4}\) the size of \(f(-2)\).

Step 3: Examine the behavior of each function as x approaches infinity

As \(x\) approaches infinity, both \(f(x)\) and \(g(x)\) approach zero. However, \(g(x)\) gets closer to zero more slowly than \(f(x)\) because it's only a quarter of \(f(x)\).

Step 4: Examine the behavior of each function as x approaches negative infinity

As \(x\) approaches negative infinity, both \(f(x)\) and \(g(x)\) approach zero. But again, \(g(x)\) approaches zero more slowly than \(f(x)\) because it's only a quarter of \(f(x)\).

Key Concepts

Function Comparison

Comparing the graphs of two functions involves examining their behaviors under different conditions. This could include evaluating their outputs for specific values or checking their reactions as the input takes on extreme values. Here, we explore the functions \(f(x) = \frac{8}{x^3}\) and \(g(x) = \frac{2}{x^3}\), where \(g(x)\) is essentially one-fourth of \(f(x)\).

To compare these two functions, you start by substituting the same values of \(x\) into both functions. For example:

• At \(x = 2\), we compute \(f(2) = 1\) and \(g(2) = 0.25\).
• At \(x = -2\), \(f(-2) = -1\) and \(g(-2) = -0.25\).

These evaluations show that \(g(x)\) is consistently a quarter of \(f(x)\) for any \(x\) value. This consistent factor means that on a graph, \(g(x)\) will always be closer to the x-axis than \(f(x)\) because it's scaled down.

The important takeaway is that when two functions differ by a constant multiple, their graphs will be similarly shaped but shifted vertically relative to each other by that consistent ratio.

Asymptotic Behavior

Asymptotic behavior refers to how functions behave as the input value \(x\) becomes very large or very small. Understanding this behavior helps in predicting how the functions will act near infinity or negative infinity. Both functions \(f(x) = \frac{8}{x^3}\) and \(g(x) = \frac{2}{x^3}\) have similar asymptotic behaviors because they are both rational functions with the same denominator.

As \(x\) approaches infinity or negative infinity, the output of both functions will approach zero. This is because, in rational functions where the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, the function tends towards zero.

However, because \(g(x)\) has a numerator one-fourth the size of \(f(x)\), it gets closer to zero more slowly than \(f(x)\). This means that \(g(x)\) flattens out toward the x-axis more gradually compared to \(f(x)\). Understanding asymptotic behavior is crucial for analyzing the end behaviors of functions, which is especially useful in calculus and graph analysis.

Rational Functions

Rational functions are quotients of two polynomial functions. The general form of a rational function is \(R(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. In our case, \(f(x) = \frac{8}{x^3}\) and \(g(x) = \frac{2}{x^3}\) are both rational functions because they are ratios of constant numerators and cubic denominators.

Important characteristics of rational functions include their vertical and horizontal asymptotes. The vertical asymptotes occur where the denominator is zero, though in our cases, \(x^3\) can only equal zero when \(x = 0\), which doesn’t form a vertical asymptote because of a zero numerator.

Horizontal asymptotes are also key, and for \(f(x)\) and \(g(x)\), they both have a horizontal asymptote at \(y = 0\). This stems from the fact that as \(x\) becomes very large or very small, the fractions \(\frac{8}{x^3}\) and \(\frac{2}{x^3}\) tend towards zero. In conclusion, by understanding these basic features of rational functions, students can better predict and analyze their graphs and behaviors.