Question

Find k that value of the function in the interval is continuous

f(x)= k cos x, x<=0 

2^x -k, x>0

Answer 1

f(x) \(=\begin{cases}k\,cos\,x;&x\leq0\\2^x-k;&x>0\end{cases}\)

f(0^-) \(=\underset{x\rightarrow0}{Lim}\) k cos x = k cos 0 = k

f(0⁺) \(=\underset{x\rightarrow0}{Lim}\) 2^x - k = 2⁰ - k = 1 - k

∵ f(x) is continuous at x = 0

∴ f(0^-) = f(0⁺)

⇒ k = 1 - k

⇒ 2k = 1

⇒ k = 1/2.