Problem 21 - College Algebra with Corequisite Support

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 function has no x- or y-intercepts because it is undefined at \(x = 0\).

Step 1: Find the x-intercept

To find the x-intercept of the function, set the numerator equal to zero and solve for \(x\). The numerator is \(x\). So, we set \(x=0\) and solve.\[ x = 0 \]Thus, the x-intercept is at \((0, 0)\).

Step 2: Determine the domain restrictions

The function has a denominator \(x^2 - x\). We need to find when this equals zero to determine any restrictions on the domain. Set \(x^2 - x = 0\):\[ x(x - 1) = 0 \]The solutions are \(x = 0\) and \(x = 1\). The domain restrictions are that the function is undefined at \(x = 0\) and \(x = 1\). Note that \((0, 0)\) is not part of the function as the function is undefined at \(x=0\).

Step 3: Find the y-intercept

To find the y-intercept of the function, evaluate the function at \(x=0\). However, since we have determined that \(x=0\) is restricted, the function does not have a y-intercept. In the context of functions defined for all \(x\), no y-intercept exists if \(x=0\) is not in the domain.

Key Concepts

X-intercept

Finding the x-intercept involves identifying a point where the graph of the function crosses the x-axis. At this point, the value of the function is zero, meaning that for any function like \( f(x) = \frac{x}{x^2 - x} \), we set the numerator, \( x \), equal to zero and solve for \( x \).

• For this function, setting \( x = 0 \) gives us an x-intercept of \( (0, 0) \).
• This matches the requirement that at the x-intercept, \( y = 0 \).

However, keep in mind, although algebra suggests that (0, 0) is an x-intercept, domain restrictions may affect whether it's part of the function, such as in this case, where \( x = 0 \) leads to an undefined function.

Y-intercept

The y-intercept is the point where a graph crosses the y-axis. To find it, you substitute \( x = 0 \) into the function and solve for \( y \), or \( f(x) \). For the function \( f(x) = \frac{x}{x^2 - x} \), substituting \( x = 0 \) would normally give you the y-intercept.

• However, since \( x = 0 \) results in an undefined function, there is no y-intercept.
• The absence of a y-intercept occurs whenever \( x = 0 \) is not within the domain of the function.

Thus, it’s crucial to check for domain restrictions before concluding the existence of a y-intercept.

Domain restrictions

Domain restrictions define values of \( x \) for which a function is undefined. In rational functions like \( f(x) = \frac{x}{x^2 - x} \), domain restrictions often occur when the denominator equals zero, as division by zero is undefined.

• To find domain restrictions, set \( x^2 - x = 0 \), and solve for \( x \).
• This gives \( x(x-1) = 0 \), leading to solutions \( x = 0 \) and \( x = 1 \).
• Thus, \( x = 0 \) and \( x = 1 \) are the domain restrictions, meaning the function is undefined at these points.

Understanding domain restrictions can clarify why sometimes a potential intercept, like an x- or y-intercept, might not be valid if it falls on an undefined point in the domain.