Question

Find a constant 'k' such that..

Answer 1

...the function is continuous.
\[f(x)\]
\[=3x+2\] for \[x <2\]
\[=x^{2}+k \] for \[x \ge2\]

Answer 2

Well, let's find the limit the 'lower' half of the function (x < 2) is approaching when x approaches 2:
$$
\lim_{x \to 2^-} 3x + 2 = 3 \cdot 2 + 2 = 6 + 2 = 8
$$
Means, the function gets closer and closer to 8 as x grows closer and closer to 2.
So we need our point at x=2 to be 8.
$$
f(2) = 8 \\
2^2 + k = 8 \\
4 + k = 8 \\
k = 4
$$

Answer 3

That's it? Thank you.

Answer 4

Yes, you're welcome