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Chapter 10: Problem 43

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

Short Answer

Expert verified
Vertical asymptote: x=0. Horizontal asymptote: y=0. x-intercept: (-1, 0). No y-intercept.

Step by step solution

01

- Find Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero, and the numerator is non-zero. For the function \(f(x)=\frac{x+1}{x^{2}}\), set the denominator equal to zero: \[ x^2 = 0 \]Solving this, we get: \[ x = 0 \]So, the vertical asymptote is at \(x = 0\).
02

- Find Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the polynomial in the numerator and the denominator. The degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.
03

- Find x-intercepts

The x-intercepts occur where the numerator is zero. For \(f(x)=\frac{x+1}{x^{2}}\), set the numerator equal to zero: \[ x + 1 = 0 \]Solving this, we get: \[ x = -1 \]So, the x-intercept is \( (-1, 0) \).
04

- Find y-intercepts

The y-intercepts occur where the function intersects the y-axis, which is when x = 0. However, for \(f(x)=\frac{x+1}{x^{2}}\), the function is undefined at x = 0 (vertical asymptote). Hence, there is no y-intercept.
05

- Sketch the Function

To sketch the function \(f(x)=\frac{x+1}{x^{2}}\), plot the x-intercept at (-1, 0). Add the vertical asymptote at x = 0, and the horizontal asymptote at y = 0. Observe the behavior of the function around these asymptotes. The function approaches these lines but never touches them.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Asymptotes
Vertical asymptotes are vertical lines where a rational function tends to infinity (positive or negative) as the input approaches a certain value. For the function \(f(x)=\frac{x+1}{x^{2}}\), we need to find where the denominator is zero. Instruction: Set the denominator equal to zero and solve for x. In this case, \(x^2 = 0\), so \(x = 0\). This tells us that there is a vertical asymptote at \(x = 0\). When sketching the graph, draw a dashed vertical line at \(x = 0\). It helps show where the function shoots to infinity.
Horizontal Asymptotes
Horizontal asymptotes indicate the value that a function approaches as the input becomes very large (positive or negative). To find these, compare the degrees of the numerator and the denominator of the rational function. In \(f(x) = \frac{x+1}{x^2}\), the degree of the numerator is 1 and the degree of the denominator is 2. Since the degree of the denominator is higher, the horizontal asymptote is at \(y = 0\). This means the function approaches the x-axis as \(x\) gets very large or very small. When sketching the graph, add a dashed horizontal line along \(y = 0\) to represent this.
x-intercepts
The x-intercepts of a function are the points where the graph crosses the x-axis. These occur where the numerator of the rational function is zero, provided the corresponding point does not make the denominator zero. For \(f(x) = \frac{x+1}{x^2}\), set the numerator equal to zero: \(x + 1 = 0\). Solving this, we find \(x = -1\). Therefore, there is an x-intercept at the point \((-1, 0)\). Include this point when sketching the graph.
y-intercepts
The y-intercepts of a function are the points where the graph crosses the y-axis. These occur at \(x = 0\). For \(f(x) = \frac{x+1}{x^2}\), substitute \(x = 0\) into the function. However, since the denominator becomes zero at \(x = 0\), the function is undefined here. Therefore, \(f(x)\) has no y-intercept. Remember this when sketching the graph and note that there鈥檚 no point where the graph crosses the y-axis.
Graph Sketching
To sketch the graph of the function \(f(x) = \frac{x+1}{x^2}\), follow these steps:
  • Mark the vertical asymptote at \(x = 0\) with a dashed line.
  • Mark the horizontal asymptote at \(y = 0\) with a dashed line.
  • Plot the x-intercept at \((-1, 0)\).
Next, observe the behavior of the function around these asymptotes. The function will approach the vertical asymptote from both sides and either rise to infinity or fall to negative infinity. Similarly, as \(x\) approaches positive or negative infinity, the function value will approach zero (the horizontal asymptote). These observations are crucial for understanding the overall shape and behavior of the graph.

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Most popular questions from this chapter

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