Problem 9 - Algebra and Trigonometry

Question

Find the \(x\) -and \(y\) -intercepts of the rational function. \(r(x)=\frac{x^{2}-9}{x^{2}}\)

Step-by-Step Solution

Answer

The x-intercepts are at \(x=3\) and \(x=-3\); there is no y-intercept.

Step 1: Find the x-intercepts

The x-intercepts of a function occur where the function equals zero, i.e., where the numerator of the rational function is zero. In this case, we need to solve the equation \(x^2 - 9 = 0\). Factoring the expression, we have \((x-3)(x+3) = 0\). Setting each factor to zero, we find the x-intercepts at \(x = 3\) and \(x = -3\).

Step 2: Find the y-intercept

The y-intercept occurs where \(x = 0\). Plug \(x = 0\) into the function \(r(x) = \frac{x^2 - 9}{x^2}\) to find the y-value. Substitute 0 into the function to get \(r(0) = \frac{0^2 - 9}{0^2}\), which is undefined, meaning there is no y-intercept for this function.

Step 3: Determine domain restrictions

Before concluding, identify restrictions. The denominator \(x^2\) cannot be zero, so \(x eq 0\). This confirms the non-existence of the y-intercept and provides critical values for asymptotic behavior.

Key Concepts

Understanding x-intercepts in Rational Functions

The x-intercepts of a rational function are those values of \( x \) where the function equals zero. In a rational function, which is a fraction of polynomial expressions, the function equals zero when the numerator is zero. Therefore, to find the x-intercepts, we specifically solve the equation where the numerator is zero.
For the rational function given as \( r(x) = \frac{x^2 - 9}{x^2} \), the numerator is \( x^2 - 9 \). Setting this equal to zero, we solve:

• \( x^2 - 9 = 0 \)
• Factor it to \( (x - 3)(x + 3) = 0 \)
• This gives us \( x = 3 \) and \( x = -3 \)

These points, \( x = 3 \) and \( x = -3 \), are where the graph of the function crosses the x-axis, known as the x-intercepts.

Exploring y-intercepts in Rational Functions

The y-intercept of a function is found where the graph crosses the y-axis. This occurs when \( x = 0 \). For a rational function, you substitute \( x = 0 \) into the function to determine the y-intercept. However, in some cases, like with the function \( r(x) = \frac{x^2 - 9}{x^2} \), the situation is different.
When substituting \( x = 0 \) in this function, we see that:

• \( r(0) = \frac{0^2 - 9}{0^2} \)
• This becomes \( \frac{-9}{0} \), which is undefined.

Since division by zero is not possible, there is no y-intercept for this rational function. This means the graph does not intersect the y-axis, highlighting an important characteristic of this function when evaluated at \( x = 0 \).

Domain Restrictions in Rational Functions

Domain restrictions determine the set of permissible \( x \) values for a function. These are critical in rational functions due to their denominators. A rational function is undefined where its denominator equals zero, as division by zero is mathematically invalid.
For \( r(x) = \frac{x^2 - 9}{x^2} \), the denominator is \( x^2 \). Setting this equal to zero, we find that:

• \( x^2 = 0 \)
• This simplifies to \( x = 0 \)

Hence, \( x = 0 \) is not in the domain of the function, meaning \( r(x) \) is undefined at this value. This not only affects the search for the y-intercept, confirming its non-existence, but it also signals an important aspect of the graph such as potential vertical asymptotes or points of discontinuity.