Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{3-x}{2-x}$$

Step-by-Step Solution

Answer

This graph for g(x) has the x-intercept at \(x = 3\), the y-intercept at \(y = -3/2\), the vertical asymptote at \(x = 2\), and the horizontal asymptote at \(y = 1\). The graph is not symmetrical.

Step 1: Finding x and y intercepts

The x-intercept is found by setting the numerator equal to zero and solving for \(x\), so if \(3 - x = 0\), then \(x = 3\). The y-intercept is found by setting \(x = 0\), so \(f(0) = -3/2\).

Step 2: Checking for Symmetry

The symmetry of a graph can be defined as either odd, even, or neither. An odd function is symmetric about the origin, while an even function is symmetric about the y-axis. To check for symmetry, replace \(x\) by \(-x\) and simplify: \(f(-x) = \frac{3-(-x)}{2 – (-x)}\). As \(f(-x)\) does not yield back to the original function \(f(x)\) or its inverse (-f(x)), therefore, this graph is not symmetrical about the origin or y-axis.

Step 3: Finding the vertical asymptote

Vertical asymptotes occur at the values of \(x\) that make the denominator zero. To find the vertical asymptotes, equate the denominator to zero and solve for \(x\), so if \(2 – x = 0\), then \(x = 2\). Therefore, the vertical asymptote is \(x = 2\).

Step 4: Finding the horizontal asymptote

Horizontal asymptotes are determined by the degree of the numerator and denominator. If the degrees are the same, the horizontal asymptote will be ratio of leading coefficients. Since in this case the degree of the numerator and denominator is the same and both leading coefficients are 1, the horizontal asymptote is \(y = 1\).

Key Concepts

X-intercept

The x-intercept of a graph is where it crosses the x-axis. To find this point for the rational function
\( f(x) = \frac{3-x}{2-x} \),
we look for values of \( x \) that make the numerator zero. For our function, this occurs when \( 3 - x = 0 \), or \( x = 3 \). Thus, the graph crosses the x-axis at the point (3, 0).
Finding the x-intercept is crucial as it gives us an anchor point for sketching the graph.

Y-intercept

Conversely, the y-intercept is where the graph hits the y-axis. This is found by evaluating the function when \( x = 0 \).
So, \( f(0) = \frac{3-0}{2-0} = -\frac{3}{2} \).
The y-intercept of our graph is (0, -1.5). It's important for sketching the overall shape and position of the function on the cartesian plane.

Graph Symmetry

Graph symmetry is a visually appealing property that can simplify the analysis and sketching of functions. There are two main types of symmetry to look for:

1. Even symmetry (symmetry about the y-axis): Occurs when \( f(x) = f(-x) \).
2. Odd symmetry (symmetry about the origin): Occurs when \( f(-x) = -f(x) \).

For the given function \( f(x) = \frac{3-x}{2-x} \), we test for symmetry by replacing \( x \) with \( -x \), resulting in a new function that does not match the original or its negative. Hence, the function has neither even nor odd symmetry. Recognizing symmetry, or the lack thereof, assists with understanding the behavior of the function across the axes.

Vertical Asymptotes

Vertical asymptotes are vertical lines that the function approaches but never touches or crosses. For the rational function \( f(x) = \frac{3-x}{2-x} \), vertical asymptotes occur where the denominator is zero. Setting
\( 2 - x = 0 \),
we find \( x = 2 \) as the vertical asymptote. This means as \( x \) approaches 2, the function's value grows without bounds. Vertical asymptotes are crucial for visualizing how the function behaves near specific values of \( x \).

Horizontal Asymptotes

Horizontal asymptotes reflect the function's behavior at extreme values of \( x \). When the degrees of the numerator and denominator are equal, like in \( f(x) = \frac{3-x}{2-x} \), the horizontal asymptote is the ratio of their leading coefficients. Both coefficients are 1 in this function, indicating a horizontal asymptote at
\( y = 1 \).
As \( x \) becomes very large or very negative, the graph of the function will get closer and closer to the line \( y = 1 \) but will not cross it. This aspect provides a sense of direction for the behavior of the function as it moves away from the origin.