X-Intercept - Algebra-2

1. Fundamental Concepts

Definition: The x-intercept of a rational function is the value of $x$ where the function intersects the x-axis, i.e., where $f(x) = 0$.
Finding X-Intercepts: To find the x-intercept, set the numerator equal to zero and solve for $x$, provided the denominator is not zero at that point.
Graphical Interpretation: On a graph, the x-intercept is the point where the curve crosses the x-axis.

2. Key Concepts

Basic Rule: $$f(x) = \frac{p(x)}{q(x)} = 0 \implies p(x) = 0$$

Degree Preservation: The degree of the polynomial in the numerator affects the number of possible x-intercepts.

Application: Used to analyze the behavior of rational functions and their graphs.

3. Examples

Example 1 (Basic)

Problem: Find the x-intercept(s) of the function $f(x) = \frac{x^2 - 4}{x + 2}$.

Step-by-Step Solution:

Set the numerator equal to zero: $x^2 - 4 = 0$
Solve for $x$: $x^2 - 4 = (x - 2)(x + 2) = 0$. Thus, $x = 2$ or $x = -2$.
Check if these values make the denominator zero: For $x = -2$, the denominator is zero, so it is not an x-intercept. For $x = 2$, the denominator is not zero.
The x-intercept is $x = 2$.

Validation: Substitute $x = 2$ into the original function: $f(2) = \frac{2^2 - 4}{2 + 2} = \frac{4 - 4}{4} = 0$. ✓

Example 2 (Intermediate)

Problem: Find the x-intercept(s) of the function $f(x) = \frac{x^3 - 8}{x^2 - 4}$.

Step-by-Step Solution:

Set the numerator equal to zero: $x^3 - 8 = 0$
Solve for $x$: $x^3 - 8 = (x - 2)(x^2 + 2x + 4) = 0$. Thus, $x = 2$.
Check if this value makes the denominator zero: For $x = 2$, the denominator is $2^2 - 4 = 0$, so it is not an x-intercept.
No valid x-intercepts exist.

Validation: Substitute $x = 2$ into the original function: $f(2) = \frac{2^3 - 8}{2^2 - 4} = \frac{8 - 8}{4 - 4} = \frac{0}{0}$, which is undefined. ✓

4. Problem-Solving Techniques

Factorization Strategy: Always factor the numerator and the denominator to simplify the function and identify potential x-intercepts.
Error-Proofing: After finding potential x-intercepts, always check that they do not make the denominator zero.
Concept Reinforcement: Use graphical representations to visualize the function and its intercepts with the x-axis.