Problem 475 - Precalculus

Question

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

Step-by-Step Solution

Answer

The x-intercept and y-intercept are both at (0,0).

Step 1: Finding the x-intercepts

To find the x-intercepts of the function, we set \(f(x) = 0\). This implies that the numerator must be zero. Thus, \(x = 0\). Therefore, the x-intercept is at \((0,0)\).

Step 2: Finding the y-intercept

To find the y-intercept, we substitute \(x = 0\) into the function. This gives \(f(0) = \frac{0}{0^2-0} = 0\). Therefore, the y-intercept is at \((0,0)\).

Step 3: Conclusion on Intercepts

Both the x-intercept and y-intercept occur at the same point \((0,0)\) because when substituting \(x = 0\) the value of \(f(x)\) is \(0\).

Key Concepts

Understanding the x-intercept

The x-intercept of a function is the point where the graph of the function crosses the x-axis. At this point, the value of the function, or the y-coordinate, is zero. To find the x-intercept, you need to set the function equal to zero and solve for the variable x.
In this exercise, the function given is \( f(x) = \frac{x}{x^2-x} \). By setting \( f(x) = 0 \), we focus only on the numerator part, since for a fraction to equal zero, its numerator must be zero while its denominator is non-zero. This simplifies to \( x = 0 \). Setting \( x = 0 \) in the function confirms no issues in division, ensuring it is a valid intercept without undefined expressions.

• The x-intercept here is \( (0, 0) \), indicating that the graph touches or crosses the x-axis at the origin.

Exploring the y-intercept

The y-intercept is where the graph crosses the y-axis. At this intersection, the value of x is zero, and the function gives its y-coordinate. To find the y-intercept, substitute \( x=0 \) into the equation and evaluate the result.
Given the function \( f(x) = \frac{x}{x^2-x} \), substituting \( x=0 \) results in \( f(0) = \frac{0}{0^2-0} = 0 \). The value is computed straightforwardly when zero is in the numerator, indicating the function's output is zero, assuming the denominator is non-zero. If both the numerator and denominator were zero, we'd explore whether the point is actually a hole or an uninterrupted point on the graph.

• The y-intercept coincides with the x-intercept in this case, since it is also \( (0, 0) \).

Diving into function analysis

Function analysis involves examining various aspects of a given function to better understand its behavior. For the function \( f(x) = \frac{x}{x^2-x} \), this involves analyzing intercepts, domain restrictions, and possible asymptotic behavior.

• Intercepts: As discussed, both the x-intercept and y-intercept occur at \( (0, 0) \).
• Domain: The domain of a function includes all x-values for which the function is defined. Here, the denominator \( x^2-x \) cannot be zero, so solving \( x^2-x=0 \) gives \( x(x-1) = 0 \), resulting in x-values of 0 and 1 where the function cannot be defined, reflecting on the graph.
• Asymptotes and Behavior: When x approaches the excluded domain values, behavior near these points may show vertical asymptotes, indicated by undefined values or tendencies toward infinity.

When analyzing, remember that understanding intercepts provides insight into where the graph intersects axes, critical in sketching and interpreting graphs.