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Identify which of the following are rational functions and which are not.

$f$ defined by $f(x)=\frac{25x^2+3}{25x^2+3}$ a rational function.

$Q$ defined by $Q(x)=\frac{5x^2+3\sqrt{x}}{2x}$ a rational function.

$g$ defined by $g(t)=\frac{t\sqrt{5} -t^3}{2t+1}$ a rational function.

$P$ defined by $P(x)=\frac{5x+3}{\abs{2x+1}}$ a rational function.

$h$ defined by $h(x)=\frac{3^x+1}{x^2+1}$ a rational function.

Solution

$f$ defined by $f(x)=\frac{25x^2+3}{25x^2+3}$ is a rational function as its formula is a polynomial divided by another polynomial.

$Q$ defined by $Q(x)=\frac{5x^2+3\sqrt{x}}{2x}$ is not a rational function because the numerator contains $\sqrt{x}$ and is therefore not a polynomial.

$g$ defined by $g(t)=\frac{t\sqrt{5} -t^3}{2t+1}$ is a rational function as its formula is a polynomial divided by another polynomial. Note that $\sqrt{5}$ is simply a real number coefficient, which is allowed in polynomial expressions.

$P$ defined by $P(x)=\frac{5x+3}{\abs{2x+1}}$ is not a rational function because the denominator contains the absolute value of an expression with variables in it, which is not a polynomial expression.

$h$ defined by $h(x)=\frac{3^x+1}{x^2+1}$ is not a rational function because the numerator contains $3^x\text{,}$ which has a variable in the exponent and thus cannot be a polynomial expression.