Problem

Which of the following are rational functions? Select all that apply.
\[
\begin{array}{l}
\square f(x)=\frac{x^{2}+5}{x} \\
\square f(x)=\frac{2^{x}}{5} \\
\square f(x)=\frac{x^{5}-x^{3}+4}{x^{2}+15 x-2} \\
\square f(x)=\frac{1}{x} \\
\square f(x)=\frac{\log (x)}{2 x+5}
\end{array}
\]

Answer

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Answer

Final Answer: The rational functions are \(\boxed{f(x)=\frac{x^{2}+5}{x}, f(x)=\frac{x^{5}-x^{3}+4}{x^{2}+15 x-2}, f(x)=\frac{1}{x}}\)

Steps

Step 1 :A rational function is a function that can be written as the ratio of two polynomials. The numerator and the denominator are both polynomials.

Step 2 :Let's check each function one by one.

Step 3 :For the function \(f(x)=\frac{x^{2}+5}{x}\), both the numerator and the denominator are polynomials, so it is a rational function.

Step 4 :For the function \(f(x)=\frac{2^{x}}{5}\), the numerator is not a polynomial, so it is not a rational function.

Step 5 :For the function \(f(x)=\frac{x^{5}-x^{3}+4}{x^{2}+15 x-2}\), both the numerator and the denominator are polynomials, so it is a rational function.

Step 6 :For the function \(f(x)=\frac{1}{x}\), both the numerator and the denominator are polynomials, so it is a rational function.

Step 7 :For the function \(f(x)=\frac{\log (x)}{2 x+5}\), the numerator is not a polynomial, so it is not a rational function.

Step 8 :Final Answer: The rational functions are \(\boxed{f(x)=\frac{x^{2}+5}{x}, f(x)=\frac{x^{5}-x^{3}+4}{x^{2}+15 x-2}, f(x)=\frac{1}{x}}\)

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