Definition of Function Notation

Consider the relation $y=3x+2$, which is a function.
The $y$-values are determined from the $x$-values, so we say '$y$ is a function of $x$, which is abbreviated to $y=f(x)$.
So, the rule $y=3x+2$ can be also be written as following. $$f: \mapsto 3x+2$$ $$\text{or}$$ $$f(x)=3x+2$$ $$\text{or}$$ $$y=3x+2$$ Function $f$ such that $x$ is converted into $3x+2$.

Example 1

If $f(x)=4x-5$, find $f(2)$.

Example 2

If $f(x)=x^2-5x+1$, find $f(-1)$.

Example 3

If $f(x)=x^2-3x+2$, find $f(x+1)$.

Example 4

Given $f(x)=ax+b$, $f(1)=7$ and $f(2)=11$, find $a$ and $b$.