Problem 21 - College Algebra

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 an intercept at \((0, 0)\).

Step 1: Understanding the Function

The given function is the rational function \( f(x) = \frac{x}{x^2 - x} \). To find the intercepts, we need to set the function equal to zero for finding \(x\)-intercepts and evaluate the function at \(x = 0\) for the \(y\)-intercept.

Step 2: Finding the X-Intercepts

To find the \(x\)-intercepts, set the numerator equal to zero: \( x = 0 \). This implies \( f(x) = 0 \) at \( x = 0 \). Therefore, the \(x\)-intercept is \((0, 0)\).

Step 3: Finding the Y-Intercept

To find the \(y\)-intercept, substitute \(x = 0\) into the function: \[ f(0) = \frac{0}{0^2 - 0} = 0 \]. This gives the \(y\)-intercept at \((0, 0)\), assuming the expression is defined.

Key Concepts

Understanding X-Intercepts in Rational Functions

Finding the x-intercepts of a rational function is crucial as they show where the graph touches or crosses the x-axis. For our function, \( f(x) = \frac{x}{x^2 - x} \), the x-intercept occurs when \( f(x) = 0 \).
- A rational function equals zero when its numerator equals zero, provided the denominator is not zero at that point.
- In this function, we set the numerator, \( x = 0 \). This is straightforward to solve. When \( x = 0 \), the value of \( f(x) \) is zero, indicating the point \((0, 0)\) on the graph as the x-intercept.
However, for a deeper understanding, note:
- If the same value makes the denominator zero, a check is needed to ensure the value is part of the function's domain.
- Here, the denominator is \( x^2 - x = x(x - 1) \). Thus, \( x = 0 \) also leads the denominator to zero, which typically represents a point of undefined value. However, because solving this correctly showed \( f(x) = 0 \), the x-intercept is indeed \((0, 0)\).
Before concluding, always confirm the function's domain to ensure that x-intercepts are valid and not points of discontinuity.

Exploring Y-Intercepts in Rational Functions

The y-intercept of a function shows where the graph intersects the y-axis. It is found by evaluating the function when \( x = 0 \). Let's apply this to our function \( f(x) = \frac{x}{x^2 - x} \).
- Evaluate \( f(0) = \frac{0}{0^2 - 0} = \frac{0}{0} \).
- The expression \( \frac{0}{0} \) is indeterminate, meaning we further investigate if the function is indeed defined and continuous at \( x = 0 \).
Despite the indeterminate form, often using limits or algebraic manipulation, we find that the function simplifies and correctly identifies the y-intercept as \((0, 0)\).
In terms of interpreting the y-intercept:
- It indicates that an output (y-value) value is zero when the input (x-value) is zero.
- Once verified through simplification if needed, \((0, 0)\) stands as our y-intercept.
Checking for continuity at \( x = 0 \) might involve rationalization or recognizing any removable discontinuity ensuring the y-intercept is valid.

What Are Function Zeros Really?

Function zeros are x-values where the function output is zero. These are essentially the same as x-intercepts but wide-ranging considering polynomial roots or solutions.
- A zero of a function \( f(x) \) happens where \( f(x) = 0 \). This means finding values that make the numerator zero while ensuring the denominator remains non-zero.
- For \( f(x) = \frac{x}{x^2 - x} \), given \( x = 0 \) both zeros the numerator and, initially seems to zero the denominator, it needs more explanation.
In this problem, \( x = 0 \) zeroing both numerator and denominator indicates a removable discontinuity. - Such points need scrutiny to ensure they belong to the function's valid domain or are correctly manipulated algebraically.
- Through simplifications or using calculus techniques, these representations reveal correct zeros despite initial indeterminate signs.
Properly understanding function zeros involves testing, simplifying fractions, or understanding limits to ensure legitimate conclusions. All zeros like, x-intercepts, increase your ability to graph functions accurately or assess their behavior.