Question

Domain Find the domain of the function.

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

Short answer

The domain of the given function is $\left(0.5,\infty \right)$.

Step-by-step solution

Step 1. Definition of domain.

The set of all input values of a function is known as the domain of the function.

Step 2. Given information.

The given function is:

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

Step 3. Domain of the given function.

The given function is a rational function. The numerator is ${\left(x+1\right)}^2$. It is defined for all real values of x.

Denominator is defined if the radicand is non negative, i.e.,$2x-1\ge 0$ but the denominator cannot be equal to $0$, therefore the function is defined if:

$$\begin{gathered} 2x-1>0 \\ 2x>1 \\ x>\frac{1}{2} \\ x>0.5 \end{gathered}$$

The given function is defined for all real values of$x$ that are greater than $0.5$.

Thus, the domain of the given function is $\left(0.5,\infty \right)$.