Find the value(s) of the constant k that make the function continuous everywhere.

Question

Find the value(s) of the constant k that make the function continuous everywhere.

f(x)= {x+k x <= 5

kx^2 x> -5

Answer 1

If you want the function to be continuous everywhere, consider that y = x + k is a line so is obviously is continuous for all values of "k" and y = kx² is a parabola which is also continuous everywhere for all values of "k". So the only place that could be an issue is at the boundary between the 2 pieces of the function.

To make it continuous, the function value of y = x + k MUST be the same as the function value of y = kx² at x = 5 (the boundary between the two pieces.

So x + k MUST be equal to kx²:

and since x = 5 then:

5 + k = 25k

5 = 24k

k = 5/24