Problem 47 - Algebra for College Students

Question

Find all asymptotes, \(x\) -intercepts, and \(y\) -intercepts for the graph of each rational function and sketch the graph of the function. $$f(x)=\frac{x}{x^{2}-1}$$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = \text{±}1 \), horizontal asymptote at \( y = 0 \), intercepts at \( (0, 0) \).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero but the numerator is not zero. Set the denominator equal to zero and solve for x: \[ x^2 - 1 = 0 \] Solve for x: \[ x^2 = 1 \] \[ x = \text{±}1 \] So, the vertical asymptotes are at \( x = 1 \) and \( x = -1 \).

Step 2: Identify Horizontal Asymptotes

If the degrees of the numerator and the denominator are the same, the horizontal asymptote is the ratio of their leading coefficients. In this case, both the numerator and the denominator have degrees of 1 and 2, respectively. Thus, the horizontal asymptote is given by \( y = 0 \).

Step 3: Find the x-Intercepts

x-intercepts occur where the numerator is zero. Set the numerator equal to zero and solve for x: \[ x = 0 \] So, the x-intercept is at \( (0, 0) \).

Step 4: Find the y-Intercepts

y-intercepts occur where the function crosses the y-axis, which is where \( x = 0 \): \[ f(0) = \frac{0}{0^2 - 1} = 0 \] So, the y-intercept is at \( (0, 0) \).

Step 5: Sketch the Graph

Using the information found in the previous steps, sketch the graph: 1. Draw vertical asymptotes at \( x = 1 \) and \( x = -1 \).2. Draw the horizontal asymptote at \( y = 0 \) (the x-axis).3. Plot the intercept at \( (0, 0) \). The graph of the function \( f(x) = \frac{x}{x^2 - 1} \) approaches these asymptotes as \( x \) goes to positive and negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. You find vertical asymptotes by setting the denominator of a rational function equal to zero and solving for x. In our example, the function is \( f(x) = \frac{x}{x^2 - 1} \).
Set the denominator equal to zero: \( x^2 - 1 = 0 \). Solve for x: \( x^2 = 1 \), hence \( x = ±1 \).
Therefore, the vertical asymptotes are at \( x = 1 \) and \( x = -1 \). These lines represent values that \( x \) cannot take, as the function would be undefined.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of a function as \( x \) approaches infinity or negative infinity. They are found by comparing the degrees of the numerator and denominator in a rational function. Here the function is \( f(x) = \frac{x}{x^2 - 1} \).
The degree of the numerator (x) is 1, and the degree of the denominator \( (x^2-1) \) is 2. Since the degree of the denominator is higher, the horizontal asymptote is \( y = 0 \), which is the x-axis. This tells us that as \( x \) becomes very large (positive or negative), the value of \( f(x) \) approaches 0.

x-intercepts

x-intercepts are points where the graph of a function crosses the x-axis. To find x-intercepts, set the numerator of the rational function equal to zero and solve for x. For \( f(x) = \frac{x}{x^2 - 1} \), set the numerator equal to zero: \( x = 0 \).
Hence, the x-intercept is at the point \( (0,0) \). This means the function passes through the origin (0,0), where the value of the function is zero.

y-intercepts

y-intercepts are points where the graph of a function crosses the y-axis. To find y-intercepts, evaluate the function at \( x = 0 \). For the given function \( f(x) = \frac{x}{x^2 - 1} \), evaluate at \( x = 0 \):
\( f(0) = \frac{0}{0^2 - 1} = 0 \).
Thus, the y-intercept is at the point \( (0,0) \). Similar to the x-intercept, this means the function passes through the origin.

Rational Functions Graphing

Graphing rational functions involves plotting the intercepts and asymptotes, then sketching the curve based on this information. For our function \( f(x) = \frac{x}{x^2 - 1} \).
Here are the steps:
1. **Draw Vertical Asymptotes:** At \( x = 1 \) and \( x = -1 \), indicating values where the function is undefined.
2. **Draw the Horizontal Asymptote:** At \( y = 0 \) (x-axis), where the function approaches as \( x \) goes to infinity.
3. **Plot Intercepts:** Place the point \( (0,0) \).
Now, sketch the curve so that it approaches the asymptotes and passes through the intercept.
Remember that the graph will approach but never touch the asymptotes. This understanding ensures you accurately depict the rational function's behavior on the graph.