Problem 28 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

Function \(g(x) = \frac{8}{(x-3)^{3}}\) is a transformation of the function \(f(x) = \frac{8}{x^{3}}\), shifted 3 units to the right. When comparing their graphs, both have the same shape, but \(g(x)\) is located 3 units to the right on the x-axis compared to \(f(x)\).

Step 1: Understanding the basic function \(f(x)\)

Consider the function \(f(x) = \frac{8}{x^{3}}\). This is a reciprocal function, or a type of rational function. It represents a curve where for every \(x\), \(f(x)\) equals the value at \(x = \frac{1}{f(x)}\). When \(x\) gets larger or smaller, the function value becomes smaller or larger, forming a curve with an inflection point at the origin.

Step 2: Understanding the transformation

The function \(g(x) = f(x-3) = \frac{8}{(x-3)^{3}}\) is a transformation of the function \(f(x)\). The '-3' inside the function effectively shifts the graph of \(f(x)\) to the right by 3 units. Therefore, the inflection point of \(g(x)\) will be at \(x = 3\) instead of \(x = 0\).

Step 3: Comparing the graphs of \(f(x)\) and \(g(x)\)

Once we plot and compare the graphs of both, the similarities and differences are observed. Both functions have the same shape due to the preservation of the original function structure, but are located at different positions on the x-axis. \(f(x)\) has an inflection point at the origin (\(x = 0\)), while \(g(x)\) has the inflection point at \(x = 3\). These observations provide us with the comparison between the two functions.

Key Concepts

Rational Functions

Rational functions are an important category in the world of algebra. They are expressed as the ratio of two polynomials. In simpler terms, a rational function is any function that can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are both polynomials, and \( Q(x) eq 0 \).

• The domains of rational functions exclude points where the denominator \( Q(x) \) equals zero, as division by zero is undefined.
• Rational functions often have vertical asymptotes at these undefined points and horizontal or oblique asymptotes that describe end behavior.

The specific function \( f(x) = \frac{8}{x^3} \) is a rational function with a cubic denominator \( x^3 \). Its domain is all real numbers except \( x = 0 \), and it doesn't have a vertical asymptote due to the nature of this cubic form but does approach zero as \( x \) goes to infinity.

Reciprocal Functions

Reciprocal functions are a specific subset of rational functions where the numerator is usually a constant and the denominator contains the variable term. In a classic form, they appear as \( y = \frac{1}{x} \), leading to a hyperbolic shape on the graph. The reciprocal function given, \( f(x) = \frac{8}{x^3} \), shows similar behavior.

• As \( x \) increases in the positive direction, \( f(x) \) decreases, approaching zero, and vice versa as \( x \) approaches zero from the positive side.
• For \( x < 0 \), this function decreases further below zero as \( x \) decreases.
• The graph of \( f(x) \) has two branches: one in the first quadrant and the other in the third quadrant, characteristic of reciprocal functions.

Understanding these makes it easier to interpret graph transformations and their implications on the function's shape and position on the coordinate plane.

Inflection Point

An inflection point on a graph is a valuable feature. It is where the concavity of the function changes direction. For the function \( f(x) = \frac{8}{x^3} \), there is an inflection point at the origin.

• This point is significant because it illustrates where the graph shifts from being concave up to concave down (or vice versa).
• In the case of \( g(x) = \frac{8}{(x-3)^3} \), the inflection point shifts to \( x = 3 \), a result of the horizontal translation performed on the original function.

Inflection points help in understanding how transformations like horizontal shifts alter the location and properties of key features in the function’s graph. They add depth to the graphical analysis by highlighting changes in the rate at which the slope of the function increases or decreases.