Find the adjoint of $\Lambda(f)=f(|x|)$.

Question

Let $H=L^2(\mathbb{R})$. Find the adjoint of $\Lambda(f)=f(|x|)$.

Here's what I did:

We have \begin{align*} \langle \Lambda(f),g \rangle &= \int_{\mathbb{R}}f(|x|)g(x) \ dx\\ &=\int_0^{\infty} f(x)(g(x)+g(-x)) \ dx\\ &=\int_{\mathbb{R}} f(x)\Lambda^*(g) \ dx \end{align*}

where $\Lambda^*(g)=\begin{cases} g(x)+g(-x) \ &\mbox{if} \ x>0 \\ 0 &\mbox{ elsewhere}\end{cases}$.

Is my work correct?