Problem 12 - Precalculus Mathematics for Calculus

Question

Find the \(x\) - and \(y\) -intercepts of the rational function. $$s(x)=\frac{3 x}{x-5}$$

Step-by-Step Solution

Answer

The function has both x- and y-intercepts at \((0, 0)\).

Step 1: Find the x-intercept

To find the x-intercept, set the numerator equal to zero and solve for \(x\). The x-intercept occurs where \(s(x) = 0\), which implies \(x = 0\). Since the numerator is \(3x\), we set \(3x = 0\). Solving this equation gives \(x = 0\). Thus, the x-intercept is \((0, 0)\).

Step 2: Find the y-intercept

To find the y-intercept, evaluate \(s(x)\) at \(x=0\). Substitute \(x=0\) into the function: \(s(0) = \frac{3 \times 0}{0-5} = \frac{0}{-5} = 0\). Thus, the y-intercept is \((0, 0)\).

Key Concepts

Understanding X-Intercept

The x-intercept of a rational function is the point where the graph crosses the x-axis. At this point, the value of the function is zero, meaning that the output is zero. In other words, to find the x-intercept, we look for values of \(x\) that make the function equal to zero.

For a rational function like \(s(x) = \frac{3x}{x-5}\), the x-intercept is found by setting the numerator equal to zero. This is because if the numerator is zero, the entire fraction becomes zero, regardless of the denominator (as long as the denominator is not also zero). Let's break this down:

• Set the numerator equal to zero: \(3x = 0\).
• Solve for \(x\): by dividing both sides by 3, we find \(x = 0\).

Therefore, the x-intercept is at the point \((0, 0)\). This tells us that the function crosses the x-axis at \(x = 0\).

Uncovering Y-Intercept

The y-intercept of a rational function is where the graph crosses the y-axis. At this crossing point, the value of \(x\) is zero because every point on the y-axis has \(x = 0\). To find the y-intercept, we substitute 0 for \(x\) in the function and solve for \(s(x)\).

Let's apply this to the function \(s(x) = \frac{3x}{x-5}\):

• Substitute \(x = 0\) into the function: \(s(0) = \frac{3 \times 0}{0 - 5}\).
• Calculate: this simplifies to \(\frac{0}{-5} = 0\).

Thus, the y-intercept is also at the point \((0, 0)\). This means the function intersects the y-axis at the same point it intersects the x-axis.

Understanding Numerator and Denominator of Rational Functions

In rational functions, the formula is expressed as the fraction of two polynomials, specifically a numerator over a denominator. How these elements behave greatly affects the graph of the rational function.

The **Numerator**:

• Determines the x-intercepts of the function.
• If the numerator equals zero, then the function equals zero, indicating a point where it crosses or touches the x-axis.

In \(s(x) = \frac{3x}{x-5}\), the numerator is \(3x\). Setting \(3x = 0\) results in an x-intercept at \(x = 0\).The **Denominator**:

• Determines the vertical asymptotes.
• If the denominator equals zero, the function becomes undefined at that point, creating a vertical asymptote.

For example, in \(s(x)\), if \(x - 5 = 0\), then \(x = 5\) is a vertical asymptote, meaning the graph approaches but never touches or crosses the line \(x=5\).

Understanding these elements can help predict and sketch the graph of rational functions.