Problem 24 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts for the functions. $$ f(x)=\frac{94-2 x^{2}}{3 x^{2}-12} $$

Step-by-Step Solution

Answer

x-intercepts: \(x = \pm \sqrt{47}\); y-intercept: \(y = -\frac{47}{6}\).

Step 1: Find the x-intercepts

For the x-intercepts, set the numerator equal to zero and solve for \(x\). The function \(f(x) = \frac{94 - 2x^2}{3x^2 - 12}\) has an x-intercept where \(94 - 2x^2 = 0\).\[94 - 2x^2 = 0\]Solve for \(x\):\[2x^2 = 94\]\[x^2 = 47\]\[x = \pm \sqrt{47}\]

Step 2: Find the y-intercept

For the y-intercept, set \(x = 0\) and evaluate the function:\[f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12}\]\[f(0) = \frac{94}{-12}\]\[f(0) = -\frac{47}{6}\]

Key Concepts

Understanding x-intercepts

When discussing x-intercepts of a function, we are referring to the points where the graph intersects the x-axis. These points occur when the output, or value of the function, is zero. For rational functions like \[f(x) = \frac{94 - 2x^2}{3x^2 - 12},\]we find x-intercepts by setting the numerator equal to zero and solving for x. This is because the fraction itself will only be zero when the numerator is zero, assuming the denominator is not zero at those points.
To find the x-intercepts:

• Set the numerator equal to zero: \(94 - 2x^2 = 0\).
• Solve for \(x\):
• Rearrange to \(2x^2 = 94\).
• Divide by 2: \(x^2 = 47\).
• Take the square root: \(x = \pm \sqrt{47}\).

The x-intercepts of this particular function are at \(x = \sqrt{47}\) and \(x = -\sqrt{47}\). Often, these values might not be neat integers, but understanding how to derive these intercepts is crucial when analyzing the behavior of rational functions.

Exploring y-intercepts

The y-intercept of a function is the point where the graph crosses the y-axis. This is found by evaluating the function when \(x = 0\). For the rational function \[f(x) = \frac{94 - 2x^2}{3x^2 - 12},\]we substitute zero for \(x\) to determine the y-intercept.
Here is the step-by-step process:

• Substitute \(x = 0\) into the function:
• \[f(0) = \frac{94 - 2(0)^2}{3(0)^2 - 12}\]
• This simplifies to \[f(0) = \frac{94}{-12}\].
• Further reduce to find \(f(0) = -\frac{47}{6}\).

This means the graph intersects the y-axis at the point \((0, -\frac{47}{6})\). The y-intercept gives important information about the function's behavior as it approaches this axis. Understanding intercepts helps paint a clearer picture of the graph and is essential for sketching it accurately.

Delving into rational functions

Rational functions are those expressed in the form \[f(x) = \frac{P(x)}{Q(x)},\]where both \(P(x)\) and \(Q(x)\) are polynomials. They can encompass a wide range of forms and behaviors due to the possibilities in the degrees and coefficients of the polynomials involved.
Key characteristics of rational functions:

• Domain: The domain excludes values that make \(Q(x) = 0\), as division by zero is undefined.
• Asymptotes: Rational functions typically have vertical, horizontal, or oblique asymptotes. Vertical asymptotes occur where the denominator is zero (and the factor is not cancelled by the numerator), while horizontal or oblique asymptotes describe end behavior.
• Intercepts: As discussed, the x-intercepts occur when the numerator is zero, provided the denominator is non-zero, while the y-intercept is found by evaluating the function at \(x = 0\).

Understanding these elements helps in analyzing and sketching rational functions. Rational functions can model various real-world scenarios, including rates, proportions, and anything involving division of quantities.