Linear Functions

Consider the functions \(f(x) = x - 1\) and \(g(x) y = -x + a\). Find the value of the real parameter \(a\in \mathbb{R}\) which ensure that the functions have a common value at \(x = 3\), i.e. \(f(3) = g(3)\).

Consider the function \[ f(x) = [x + 2] \] defined on the domain \(\mathop{\mathrm{Dom}}(f) = (1;2)\). Find the parameters \(a\) and \(b\) in the linear function \[ g(x) = ax + b \] which ensure that the functions \(f\) and \(g\) are identical on the domain of \(f\). \[ \] Hint: The function \(y = [x]\) is a floor function: the largest integer less than or equal to \(x\). For positive \(x\) it is also called the integer part of \(x\).

Consider the functions \(f(x) = x + 1\) and \(g(x) = ax + 7\). For what values of the real parameter \(a\in \mathbb{R}\), do the functions have the same function value equal to \(3\).

The linear function \(g\) is graphed in the picture. Find the analytic expression for the function \(g\).