Question

Find the constant k that makes the piecewise defined function (defined below) continuous.
f(x) =x + k if x less than or equal to  2
5 − x if x > 2
1my anserw was k=3/2 is that right?

Answer 1

\(f(x) = x+k\) is continuous for all \(x \le 2\). Similarly, \(f(x) = 5-x\) is continuous for all \(x > 2\). The only place where the continuity might break is at the point \(x=2\).
The left hand limit of \(f(x)\) at \(x=2\) is \(LHL = f(2_{-}) = 2 + k\) and the right hand limit is \(RHL = f(2_{+}) = 5-2 = 3\)

The function is continuous in its domain if LHL = RHL \(\implies 2+k = 3\)
i.e. \({k=1}\)

Answer 2

Thank you