Think of a line with slope $L-1$. That means that any change in $x$ will get multiplied by $L-1$ to get the change in $y$. Since we want the total change in $y$ to be more than $A$, that means that the change in $x$ must be more than $A/(L-1)$.
And of course, since $L-1$ is the minimum of $f'$, the total change could of course be larger; but this is a lower bound. (Note we need $L>1$ for our choice of $\epsilon$ to work, but you can always choose an appropriate $\epsilon$, say $L/2$ for whatever $L$ happens to be.)
I see there are already a few answers to this question but I'm going to try my hand at answering it the way I see it.
A function $f$ is a mathematical object that relates elements of two sets, one called the domain $A$ and one called the codomain $B$. The notation $f : A \to B$ denotes the fact that $f$ is a function with domain $A$ and codomain $B$.
What it means to be a function $f : A \to B$ is this: $f$ assigns to each element of $A$ exactly one element of $B$. If $a \in A$, the notation $f(a)$ denotes the element of $B$ to which $a$ is assigned by $f$.
Those elements of $B$ which can be written in the form $f(a)$, for some $a \in A$, are called values of $f$. The set of all values of $f$ is the image of $f$, which is a subset of $B$ (and not necessarily all of $B$).
There are various ways of specifying functions. For example:
- If $A$ is finite, you can simply list the values of $f$. For example we can define a function $f : \{ 1,2,3 \} \to \{ \text{red}, \text{green}, \text{blue} \}$ by $$f(1) = \text{green}, \quad f(2) = \text{blue}, \quad f(3) = \text{red}$$
- Sometimes functions can be defined by an equation. An example of a function $f : \mathbb{R} \to \mathbb{R}$ is the one defined by the equation
$$f(x) = x^2+3$$
This equation is not itself a function. What it means is, given an element $x \in \mathbb{R}$, the value of $\mathbb{R}$ associated by $f$ with $x$ is $x^2+3$. The expressions $f(x)$ and $x^2+3$ both denote exactly the same thing here: the real number associated with the number $x$. For example $f(2)$ denotes the same thing as $2^2+3$, which in turn denotes the same thing as $7$. The function is $f$, and $f(x)$ denotes the value of $f$ at a given number $x$.
The graph of a function $f : A \to B$ is the set $$\{ (a,b) \in A \times B : b = f(a) \}$$
Sometimes, particularly when $f : \mathbb{R} \to \mathbb{R}$, it is convenient to define a function in terms of its graph. (Indeed, $\mathbb{R} \times \mathbb{R}$ what you're depicting when you draw a pair of coordinate axes.) For example the equation $y=x^2+3$
specifies a function $f : \mathbb{R} \to \mathbb{R}$ whose graph is the set
$$\{ (x,y) \in \mathbb{R} \times \mathbb{R} : y=x^2+3 \}$$
That is, the function $f$ specified by this equation is the one which associates to each $x \in \mathbb{R}$ the value $x^2+3$. Some people would then say '$y$ is a function of $x$', but this is slightly misleading: what it means is that there is a function whose values are exactly the values of $y$ satisfying the given equation.
An example of how a function works is as follows. Suppose a bird is flying in a straight line at a constant speed of $12$ metres per second. The distance the bird flies 'is a function of time', in the following sense: if $t$ is a positive real number, then the distance flown by the bird in $t$ seconds is $12t$ metres. Thus the relationship between distance and time defines a function $d : \mathbb{R}^+ \to \mathbb{R}$, which is defined by the equation
$$d(t) = 12t$$
for all $t \in \mathbb{R}^+$. Thus you can consider $d$ as being the 'distance function', and for each $t$ you can consider $d(t)$ as the 'distance travelled at time $t$'.
Some non-examples of functions are:
- $f : \mathbb{R} \to \mathbb{Q}$ defined by $f(x) = x$. This is not a function because, for example, $f(\sqrt{2})=\sqrt{2}$, which is not an element of $\mathbb{Q}$.
- $f : \mathbb{R}^+ \to \mathbb{R}$ defined by $f(x)^2=x$. This is not a function because, for example, the expression $f(1)$ has more than one possible value satisfying the equation, namely $1$ or $-1$.
- $f : \mathbb{R} \to \mathbb{R}$ defined by the graph $x^2+y^2=1$. This is not a function because the values of $y$ are not uniquely determined by the values of $x$, for example $0^2+1^2=1$ and $0^2+(-1)^2=1$.
In summary, if $f : A \to B$ is a function, then
- $f$ is the function itself, which has domain $A$ and codomain $B$;
- $f(a)=(\text{expression in terms of}\ a)$ is an equation which specifying $f$ by declaring its effect on the elements of $A$; the expression $f(a)$ is not itself a function ($f$ is the function), but a function is determined by its values, so specifying $f(a)$ suffices;
- $y=f(x)$ is an equation that specifies $f$ in terms of its graph.
In the mathematical branch of set theory, which is used as a foundation for most mainstream mathematics, we need to specify precisely in terms of sets what it means for $f$ to be a function. In this setting, a function $f : A \to B$ is usually defined to be its graph: that is, if $f$ assigns to each $a \in A$ the value $f(a) \in B$, then we'd write
$$f = \{ (a,b) \in A \times B : b=f(a) \} \subseteq A \times B$$
This formal approach isn't something you need to worry about if you're learning about functions for the first time. All that matters is that for every element of the domain $A$, $f$ identifies that element with exactly one element of the codomain $B$.
This explanation is woefully incomplete, but there's only so much you can do in an MSE answer... let me know if you need more clarifications.
Best Answer
You want to compute $\sup_x f(x)$ with $f : \mathbb{R}_{++} \to \mathbb{R}$ defined by $f(x) = yx+\log(x)$. For computing the supremum, it is useful to know the derivative:$$\frac{df}{dx} = y+\frac{1}{x}.$$ Since $x>0$, $df/dx>0$ when $y\geq 0$. That means that when $y\geq 0$:$$\sup_x f(x) = \lim_{x \to \infty} f(x) = \infty.$$ On the other hand, when $y<0$, the derivative is positive for $x<-1/y$, and negative for $x>-1/y$. That means that the supremum is attained at $x=-1/y$.