Problem 1 - Algebra 2

Question

Find any points of discontinuity for each rational function. $$ y=\frac{2 x^{2}+5}{x^{2}-2 x} $$

Step-by-Step Solution

Answer

The points of discontinuity for the rational function \(y=\frac{2 x^{2}+5}{x^{2}-2 x}\) are \(x = 0\) and \(x = 2\).

Step 1: Identify the denominator

The denominator of the rational function given is \(x^{2}-2x\). It's essential to find its roots to identify the points of discontinuity.

Step 2: Set the denominator equal to zero

To find the roots of the denominator \(x^{2}-2x\), set it equal to zero: \(x^{2}-2x = 0\).

Step 3: Solve for x

Factoring out an x from \(x^{2}-2x\), we get \(x(x-2) = 0\). Therefore, the roots of \(x^{2}-2x\), and thus the points of discontinuity, are \(x = 0\) and \(x = 2\).

Key Concepts

Rational Functions

Rational functions are a type of function that is expressed as the ratio of two polynomials. In other words, a rational function has the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). This distinction is crucial because the denominator cannot equal zero, as division by zero is undefined. Therefore, the values of \( x \) that make \( Q(x) = 0 \) must be identified and analyzed for points of discontinuity. These are places where the function is not defined or does not behave as expected.

Roots of a Function

The roots of a function are the values of the variable that make the function equal to zero. In the context of a rational function, if you want to find the points of discontinuity, you'll focus on the roots of the denominator. Here, finding the roots involves solving the equation \( Q(x) = 0 \). For example, in the function given \( y=\frac{2x^{2}+5}{x^{2}-2x} \), the roots of the denominator \( x^{2} - 2x \) are computed by setting \( x^{2} - 2x = 0 \) and solving for \( x \). Identifying these roots helps us understand where the function becomes undefined, creating discontinuities.

Denominator in Algebra

The denominator in algebra is the bottom part of a fraction. When dealing with rational functions, the denominator plays a critical role in determining the domain of the function. To find the points where the rational function is not defined, set the denominator equal to zero and solve for the variable. This process identifies the values which cause division by zero. In the function \( y=\frac{2x^{2}+5}{x^{2}-2x} \), the denominator \( x^{2} - 2x \) needs careful examination.

• Set \( x^{2} - 2x \) to zero to find the critical points.
• This provides crucial information for finding discontinuities.

Factoring Equations

Factoring equations is a method used to simplify expressions and solve equations, especially when looking for the roots. Factoring involves expressing a polynomial as a product of its factors. In this case, for the rational function's denominator \( x^{2} - 2x \), factoring allows us to find its roots by transforming \( x^{2} - 2x \) into \( x(x - 2) = 0 \). Once factored:

• Set each factor to zero: \( x = 0 \) and \( x - 2 = 0 \).
• This gives the solutions \( x = 0 \) and \( x = 2 \), which are the points of discontinuity.

Factoring is an essential skill in algebra for simplifying complex expressions and solving for unknown values.