Finding the domain and range of a rational function

Expert intro: This problem asks for the domain and range of a rational function after identifying where the denominator is zero and how the graph behaves as a transformed reciprocal.

Detailed walkthrough

Key idea: rational functions exclude denominator zeros

For a function like

$f(x)=\frac{2}{3x+2},$

the first step is to determine where the expression is undefined. A rational function cannot take any input that makes the denominator equal to zero, because division by zero is not allowed.

Set the denominator equal to zero:

$3x+2=0 \quad \Rightarrow \quad x=-\frac{2}{3}.$

So the domain is all real numbers except $x=-\frac{2}{3}$.

Range comes from the reciprocal structure

The function is a scaled version of the reciprocal function $p(x)=-\frac{1}{x}$. A reciprocal-type graph can never output $0$, because the numerator is a nonzero constant and the graph approaches, but never crosses, the horizontal asymptote $y=0$.

For $f(x)=\frac{2}{3x+2}$, no value of $x$ makes the output equal to zero. That means the range is all real numbers except $y=0$.

You can also see this algebraically. If

$y=\frac{2}{3x+2},$

then solving for $x$ gives

$y(3x+2)=2,$ $3xy+2y=2,$ $x=\frac{2-2y}{3y},$

which is only possible when $y\neq 0$. So $y=0$ must be excluded from the range.

Final domain and range

• Domain: $(-\infty,-\tfrac23)\cup(-\tfrac23,\infty)$
• Range: $(-\infty,0)\cup(0,\infty)$

Common mistake to avoid

A frequent error is to say the range excludes $-\frac23$ because that is the excluded input. That is not correct. The excluded input affects the domain, not the range. The output restriction comes from the reciprocal form, which excludes $0$ from the set of possible $y$-values.

💡 Pitfall guide

A common pitfall is mixing up the excluded x-value with the excluded y-value. The number $-\frac23$ is not a range restriction; it is only the input that makes the denominator zero. Another mistake is trying to infer the range from a table of sample values and missing the asymptotic behavior. For rational functions of the form $\frac{a}{bx+c}$ with $a\neq 0$, the output can never be zero, no matter how large or small $x$ becomes. Always check both the denominator and the reciprocal structure.

🔄 Real-world variant

If the function were changed to $g(x)=\frac{2}{3x+2}+5$, the domain would stay the same because the denominator is unchanged: $x\neq -\frac23$. But the range would shift upward by 5, becoming all real numbers except $y=5$. That happens because the vertical translation moves the horizontal asymptote from $y=0$ to $y=5$. A similar variant is $h(x)=\frac{-2}{3x+2}$, which still excludes $x=-\frac23$ from the domain, but the graph is reflected across the x-axis while the range still excludes $0$.

🔍 Related terms

rational function domain, horizontal asymptote, reciprocal function