Pregunta
Find the domain of the following rational function. \[ R(x)=\frac{13 x}{x+15} \]
Ask by Pollard Hodges. in the United States
Mar 15,2025
Solución de inteligencia artificial de Upstudy
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The domain of \( R(x) = \frac{13x}{x+15} \) is all real numbers except \( x = -15 \).
Solución
1. The rational function is given by
\[
R(x)=\frac{13x}{x+15}.
\]
2. The domain of a rational function includes all real numbers except those that make the denominator zero. For this function, the denominator is \(x+15\).
3. Set the denominator equal to zero and solve:
\[
x+15=0 \quad \Rightarrow \quad x=-15.
\]
4. Since \(x=-15\) makes the denominator zero, it is excluded from the domain.
5. Therefore, the domain of \(R(x)\) is all real numbers except \(-15\), which can be written in interval notation as:
\[
(-\infty, -15) \cup (-15, \infty).
\]
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Simplifique esta solución Bonus Knowledge
To find the domain of the rational function \( R(x)=\frac{13 x}{x+15} \), we need to determine where the function is defined. A rational function is undefined wherever the denominator equals zero. For this function, we set the denominator \( x + 15 = 0 \) and solve for \( x \): \[ x + 15 = 0 \implies x = -15 \] Therefore, the function \( R(x) \) is undefined at \( x = -15 \). The domain is all real numbers except this point. In interval notation, the domain of \( R(x) \) can be expressed as: \[ (-\infty, -15) \cup (-15, \infty) \] This means you can plug in any real number for \( x \) except for \( -15 \) and still get a result from \( R(x) \).
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