1) Which of the following choices are rational functions? Check all that apply.
A. $F(x)=\frac{6}{x^{2}}$
B. $F(x)=\frac{x+9}{x^{3}-4}$
C. $F(x)=\frac{3}{5 x}$
D. $F(x)=\frac{9}{x+1}$
E. $F(x)=\frac{x^{2}+\sqrt{x}}{x+7}$
\(\boxed{\text{The rational functions are A, B, C, and D. The function E is not a rational function.}}\)
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. The denominator cannot be zero.
Step 2 :Looking at the options:
Step 3 :$F(x)=\frac{6}{x^{2}}$ - This is a rational function. The numerator is a constant (which is a polynomial) and the denominator is a polynomial.
Step 4 :$F(x)=\frac{x+9}{x^{3}-4}$ - This is a rational function. Both the numerator and the denominator are polynomials.
Step 5 :$F(x)=\frac{3}{5 x}$ - This is a rational function. The numerator is a constant (which is a polynomial) and the denominator is a polynomial.
Step 6 :$F(x)=\frac{9}{x+1}$ - This is a rational function. The numerator is a constant (which is a polynomial) and the denominator is a polynomial.
Step 7 :$F(x)=\frac{x^{2}+\sqrt{x}}{x+7}$ - This is not a rational function. The numerator is not a polynomial because of the square root of x.
Step 8 :Final Answer: The rational functions are A, B, C, and D. The function E is not a rational function.
Step 9 :\(\boxed{\text{The rational functions are A, B, C, and D. The function E is not a rational function.}}\)