Problem 46 - Algebra 2

Question

Graph each rational function. $$ f(x)=\frac{x^{2}-4 x}{x-4} $$

Step-by-Step Solution

Answer

The function is a line \(y = x\) with a hole at \(x = 4\).

Step 1: Identify the Domain

Identify the values of the domain for which the function is undefined. Here, the denominator is zero when \(x = 4\). Therefore, the function is undefined at \(x = 4\). The domain of the function is all real numbers except \(x = 4\).

Step 2: Factor the Function

Factor the numerator,\(x^2 - 4x\), which is \(x(x - 4)\). Rewrite the function as \(f(x) = \frac{x(x-4)}{x-4}\).

Step 3: Simplify and Identify Holes

Cancel the \(x-4\) term in the numerator and the denominator. The simplified form is \(f(x) = x\), but since \(x = 4\) is not in the domain, there is a hole at \(x = 4\).

Step 4: Intercepts Identification

Set the simplified function \(f(x) = x\) to find intercepts. The y-intercept is at (0,0) and since the function is linear \(f(x) = x\), every intercept is on the line \(y = x\).

Step 5: Draw the Graph

Plot the line \(y = x\), and mark a hole at \(x = 4\). The graph will be a straight line with a discontinuity (hole) at \(x = 4\).

Key Concepts

Domain of a Function

Understanding the domain of a rational function is key to knowing its behavior and where it exists within the number line. The domain consists of all possible input values (x-values) that the function can accept without causing undefined behaviors or contradictions. In rational functions, this usually involves avoiding division by zero since dividing by zero is undefined.
For example, if we have a function like \[f(x) = \frac{x^2 - 4x}{x-4}\]the denominator will be zero when \(x = 4\). Therefore, the function is undefined at \(x = 4\), which means the domain excludes this value. In this case, the domain is all real numbers except \(x = 4\). Recognizing these exclusions is vital for identifying points of discontinuity in the function's graph, such as holes or vertical asymptotes.

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions, like the numerator of a rational function. It involves breaking down a complex polynomial into products of simpler polynomials.
Consider the polynomial \(x^2 - 4x\) from our function. This can be factored using common factoring techniques:

• Notice that both terms share an \(x\), so we can factor out \(x\) to get: \(x(x - 4)\).

This turns the original function\[f(x) = \frac{x^2 - 4x}{x-4}\]into a new version with visible factors:\[f(x) = \frac{x(x-4)}{x-4}\]Factoring not only helps in simplifying functions but also assists in identifying potential reductions and detecting holes. Such simplifications can make it easier to discern the function's behavior when graphing.

Intercepts of a Function

Intercepts are the points where a function crosses the axes, and they provide insight into the function's behavior. The most common intercepts are:

• Y-intercept: Where the function crosses the y-axis. This occurs when \(x = 0\).
• X-intercepts: Where the function crosses the x-axis. This occurs when \(f(x) = 0\).

For the simplified function \(f(x) = x\), the y-intercept is straightforward. Substituting \(x = 0\) gives \(f(0) = 0\), meaning the y-intercept is (0,0). For every x-value where \(f(x) = 0\), we find the x-intercepts, which in this linear form also occurs at the origin when plotted along the line \(y = x\). Understanding intercepts helps us orient the graph correctly within the coordinate plane.

Graphing Rational Functions

Graphing rational functions involves carefully plotting the simplified expression and noting any discontinuities or features like holes and asymptotes. After simplifying the function as given in\[f(x) = \frac{x(x-4)}{x-4} = x\],we identify that the simplified function is linear, represented by the equation \(y = x\).
However, the original function has a point of discontinuity at \(x = 4\) because this value was removed in simplification as \(x = 4\) is not in the domain. This creates a 'hole' in the graph, so while the line appears unbroken, we must illustrate a gap at this point.
Plotting the line involves drawing a straight line through the origin (0,0) following the pattern of \(y = x\), but remembering to visibly mark and acknowledge the hole at \(x = 4\) confirms the graph's continuity and accurate portrayal of the function. This integrated approach provides a comprehensive understanding of rational functions and their graphical representation.