Question

\(R(x)=\frac{x(x-2)^{2}}{x-2}\) (a) Find the domain of \(R\). (b) Find the \(x\) -intercepts of \(R\).

Short answer

Domain: all real numbers except \(x = 2\). \(x\)-intercept: \(x = 0\).

Step-by-step solution

Simplify the Rational Expression

First, simplify the rational expression. Notice that the factor \(x-2\) cancels out between the numerator and the denominator. Therefore, \(R(x) = x(x-2)^{2} / (x-2) = x(x-2) \). However, note that \(x eq 2\) since it would make the original denominator zero.

Find the Domain

To find the domain, identify the values of \(x\) that make the denominator zero. The original denominator is zero when \(x = 2\). Thus, all real numbers except \(x = 2\) are in the domain. So, the domain is \,\R\ \ \{2\}

Set the Numerator Equal to Zero

To find the \(x\)-intercepts, set the numerator equal to zero: \(x(x-2) = 0\). Solving this equation gives \(x = 0\) or \(x = 2\). However, since \(x = 2\) is not in the domain, the \(x\)-intercept is only at \(x = 0\).

Write the Results

The domain is all real numbers except \(x = 2\), and the only \(x\)-intercept is at \(x = 0\).

Key concepts

Domain of a Function

The domain of a function is the set of all possible input values (usually represented by x) that the function can accept without causing any mathematical errors, like division by zero. In rational functions, the domain is influenced mainly by the denominator. If the denominator is zero at any point, the function is undefined at that value.
For the given rational function, \(R(x) = \frac{x(x-2)^2}{x-2}\), let's identify its domain.
The simplified form of \(R(x)\), after canceling the common factor \(x-2\), is \(R(x) = x(x-2)\). But remember, if \(x = 2\), the original denominator \(x-2\) becomes zero, making the function undefined.
Therefore, the domain of the function \(R(x)\) is all real numbers except \(x = 2\). This can be represented as \(\text{Domain}(R) = \mathbb{R} \backslash \{2\}\).
Conclusion:

• Identify points that make the denominator zero.
• Exclude those points from the domain.

X-Intercepts

The x-intercepts of a function are the points where the function crosses the x-axis. These occur where the output value (y) is zero. To find the x-intercepts of a rational function, we set the numerator equal to zero and solve for x.
For \(R(x) = \frac{x(x-2)^2}{x-2}\), we simplify it to \(R(x) = x(x-2)\). To find the x-intercepts:
1. Set the numerator to zero: \(x(x-2) = 0\)
2. Solve the equation: \(\begin{cases} x = 0 \ x-2 = 0 \end{cases}\)
This gives us \(x = 0\) and \(x = 2\). However, since \(x = 2\) is not in the domain (as it makes the original denominator zero), \(x = 2\) cannot be an x-intercept.
Therefore, the only x-intercept is at \(x = 0\).
Key Points:

• Set the numerator equal to zero.
• Solve for x.
• Ensure solutions are within the domain.

Simplifying Rational Expressions

Simplifying rational expressions is the process of reducing them to their simplest form. This involves canceling common factors in the numerator and the denominator.
For the rational function \(R(x) = \frac{x(x-2)^2}{x-2}\), we can simplify it by canceling the common factor \(x-2\) in the numerator and the denominator:
1. Factoring common terms: \( \frac{x(x-2)^2}{x-2} \ = \frac{x(x-2)(x-2)}{x-2} \).
2. Canceling the common factor: \( \frac{x(x-2)(x-2)}{x-2} = x(x-2) \) (note \(x eq 2\) because it makes the original denominator zero).
This results in the simplified form \(R(x) = x(x-2)\).
Tips for Simplification:

• Identify and factor common terms in the numerator and denominator.
• Cancel the common factors.
• Check for any conditions that may result from the cancellation (e.g., values that make the original denominator zero).