Question

The domain of the function$f\left(x\right)=\frac{x+1}{2x+1}$is _______.

Short answer

The domain of the function$f\left(x\right)=\frac{x+1}{2x+1}$is $\left\{x|x\ne -\frac{1}{2}\right\}$.

Step-by-step solution

Step 1. Given information.

We have:

$f\left(x\right)=\frac{x+1}{2x+1}$

Step 2. Find the domain.

The domain of the function is the set of input values for which the function is real and defined.

Find the undefined points by comparing the denominator to zero:

$$\begin{gathered} 2x+1=0 \\ 2x=-1 \\ x=-\frac{1}{2} \end{gathered}$$

The function domain is $x<-\frac{1}{2}\mathrm{or}\hspace{0.17em}x>-\frac{1}{2}$.

Interval notation: $\left(-\infty \hspace{0.17em},\hspace{0.17em}-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\hspace{0.17em}\infty \hspace{0.17em}\right)$

Set builder form:$\left\{x|x\ne -\frac{1}{2}\right\}$