I am working with an exercise regarding the implicit function theorem. Given
$$F(x, y) = y(e^y + x) − \log(x),$$
I know that $F(x_0, y_0) = 0$ for $(x_0,y_0)=(1,0)$.
I want to check that there exists a strictly positive real number $\delta > 0$ and a function $y =f(x)$ such that $$y_0 = f(x_0)\quad \text{ and }\quad F(x, f(x)) = 0 \qquad \forall x ∈ (x_0 − \delta, x_0 + \delta).$$
Is it fair to say that, in order to solve the question, we have to check if it is possible to apply the first part of the Implicit Function Theorem? Which would imply, as the answer sheet says, that the point $(1, 0)$ belongs to $$DF = \{(x, y) \in \mathbb{R}^2∶ x > 0, y \in \mathbb{R}\},$$ the domain of $F$.
I have dyscalculia and it appears confusing how to solve the domain of this function, where $x$ is greater than $0$ and $y$ exists in real numbers.
Can somebody please show me, in the simplest way possible, how to find the domain so I can proceed with the excercise? Thanks!
The formula defining $F$ only makes sense for $x>0$ because pf the second term, $\log(x)$. The largest possible domain of the logarithm function is all positive real numbers, so we need $x>0$. There is no restriction on $y$ because $y(e^y+x)$ makes sense for any real number $y$ (the largest possible domain of the exponential function is all real numbers).