Problem 39 - 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. $$f(x)=\frac{1}{x^{2}}$$

Step-by-Step Solution

Answer

Vertical asymptote: x = 0 Horizontal asymptote: y = 0 No x-intercepts or y-intercepts.

Step 1: Identify vertical asymptotes

Vertical asymptotes occur where the denominator of the rational function is zero and not canceled by the numerator. For the function \[ f(x) = \frac{1}{x^2} \] set the denominator equal to zero and solve for x: \[ x^2 = 0 \] The solution is \[ x = 0 \] Thus, there is a vertical asymptote at \[ x = 0 \]

Step 2: Identify horizontal asymptotes

To determine the horizontal asymptote, examine the behavior of the function as \[ x \rightarrow \pm \infty \] Since the degree of the polynomial in the denominator (2) is greater than the degree of the polynomial in the numerator (0), the horizontal asymptote is \[ y = 0 \]

Step 3: Find the x-intercepts

The x-intercepts are found by setting the numerator equal to zero and solving for x. For \[ f(x) = \frac{1}{x^2} \] the numerator is 1, which is never zero. Therefore, there are no x-intercepts.

Step 4: Find the y-intercepts

The y-intercept is found by evaluating the function at x = 0. However, since \[ f(0) = \frac{1}{0^2} \] is undefined due to division by zero, there is no y-intercept.

Step 5: Sketch the graph

Using the determined information, sketch the graph of the function \[ f(x) = \frac{1}{x^2} \] Vertical asymptote at \[ x = 0 \] Horizontal asymptote at \[ y = 0 \] The function is always positive and tends to infinity as x approaches zero from either direction. As x approaches fty, f(x) approaches 0.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where the function grows without bound as the input approaches a specific value. For the function \[ f(x) = \frac{1}{x^2} \], we find the vertical asymptotes by setting the denominator equal to zero. Solving \[ x^2 = 0 \], we get \[ x = 0 \]. This means our function has a vertical asymptote at \[ x = 0 \]. In simpler terms, as we get closer to \[ x = 0 \] from either the left or right, the function’s value increases indefinitely.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values become very large (positively or negatively). For \[ f(x) = \frac{1}{x^2} \], as \[ x \] approaches \[ \pm \infty \], the value of \[ f(x) \] approaches \[ 0 \]. This happens because the degree of the denominator is greater than the degree of the numerator. So, the horizontal asymptote for this function is \[ y = 0 \]. In layman's terms, this means that as \[ x \] gets very large in either direction, \[ f(x) \] gets closer and closer to zero.

X-Intercepts

X-intercepts are where the function crosses the x-axis (where y=0). To find them, set the numerator of \[ f(x) = \frac{1}{x^2} \] equal to zero and solve for x. Since the numerator is \[ 1 \] and never zero, \[ f(x) \] has no x-intercepts. In simpler terms, this function will never cross the x-axis.

Y-Intercepts

Y-intercepts are where the function crosses the y-axis (where x=0). To find this for \[ f(x) = \frac{1}{x^2} \], we set \[ x \] to zero. However, plugging in \[ x = 0 \] makes the denominator zero, thus making the expression undefined. Therefore, there is no y-intercept for this function. Simply put, this function will never cross the y-axis.

Graphing Rational Functions

To graph \[ f(x) = \frac{1}{x^2} \], follow these steps:

• Identify and draw the vertical asymptote at \[ x = 0 \].
• Identify and draw the horizontal asymptote at \[ y = 0 \].
• Recognize that the function is always positive and tends to infinity as \[ x \] approaches zero from either direction.
• As \[ x \] gets very large (positively or negatively), \[ f(x) \] approaches zero.

Visualizing this, the graph will be U-shaped, facing upwards, get very high near \[ x = 0 \] and flatten out as it moves away from the y-axis. This gives a clear picture of how \[ f(x) \] behaves across different values of \[ x \].