Problem 51 - Algebra 2

Question

REASONING Compare and contrast the graphs of \(f(x)=\frac{(x-1)(x+5)}{x-1}\) and \(g(x)=x+5\)

Step-by-Step Solution

Answer

The graph of \(f(x)\) has a hole at \(x=1\), while \(g(x)\) is a continuous line.

Step 1: Analyze the function f(x)

Given the function \(f(x)=\frac{(x-1)(x+5)}{x-1}\), let's analyze its form. Notice that the \(x-1\) in the numerator and denominator can potentially cancel out, except where it makes the denominator zero. Thus, the function simplifies to \(f(x)=x+5\) except at \(x=1\) where there is a discontinuity.

Step 2: Identify the discontinuities in f(x)

In the function \(f(x)\), the denominator becomes zero when \(x=1\). Therefore, there is a point of discontinuity, specifically a removable discontinuity, at \(x=1\). The function is undefined at this point, which means there will be a hole in the graph at \(x=1\).

Step 3: Analyze the function g(x)

The function \(g(x)=x+5\) is a linear function with no discontinuities. Its graph is a straight line with a slope of 1, intercepting the y-axis at the point \((0, 5)\).

Step 4: Compare the graphs

To compare, \(f(x)\) and \(g(x)\) both yield the expression \(x+5\) for values where \(x eq 1\). Thus, on the graph, they look the same except at \(x=1\). The graph of \(f(x)\) will have a hole at \(x=1\), while the graph of \(g(x)\) will be a continuous straight line, including at \(x=1\).

Step 5: Contrast the graphs

The key difference is that \(f(x)\) has a hole at \(x=1\), whereas \(g(x)\) does not. This difference in continuity is the main contrast between these two graphs.

Key Concepts

Removable Discontinuity

A removable discontinuity in a function occurs when there is a point at which the function is not defined, even though the rest of the function behaves normally. Typically, this happens when a factor in the numerator and the denominator can cancel out, except at a specific value that would make the denominator zero. For example, in the function \(f(x)=\frac{(x-1)(x+5)}{x-1}\), the \(x-1\) terms cancel out, simplifying the expression to \(f(x)=x+5\), except when \(x=1\).
This point \(x=1\) is where the denominator becomes zero, so the function is not defined there, resulting in what's known as a "hole" in the graph at that point.
The function, elsewhere continuous, physically presents this discontinuity as just a missing single point in the otherwise smooth curve.

Linear Functions

Linear functions are one of the simplest types of functions to graph and understand. A linear function like \(g(x)=x+5\) presents a straight line when plotted on a coordinate plane. Here, the equation is of the form \(y=mx+b\),
where \(m\) represents the slope and \(b\) represents the y-intercept. In the given equation, \(m=1\) and \(b=5\), so the graph is a straight line with a slope of 1, meaning it rises one unit vertically for every one unit it moves horizontally, and crosses the y-axis at \((0, 5)\).
Linear functions can be contrasted with many other types of functions because they include no bends or curves and do not possess any discontinuities. They extend infinitely in both directions along the x-axis.

Discontinuity Analysis

Discontinuity analysis is crucial for understanding where and why a function is not defined, or not behaving in a typical continuous manner. In terms of graphing, discontinuities can manifest as jumps, infinite discontinuities, or removable points. Removable discontinuities, as seen in the function \(f(x)=\frac{(x-1)(x+5)}{x-1}\), occur when a common factor results in a simplified expression that doesn't permit the function to be defined at a particular point.
To analyze for discontinuities, examine the denominator first for zero-points. In our example, \(x=1\) causes the denominator to be zero, indicating a discontinuity there.
However, graphically comparing \(f(x)\) and \(g(x)\), the former contains the hole at \(x=1\), while \(g(x)=x+5\) remains continuous everywhere, illustrating the subtleties and importance of comprehensive analysis.