Question

for what value of the constant c is the function f continuous on (-∞, ∞) where f(t) = {t² - c if t < 8; ct + 4 if t ≥ 8}

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $t = 8$, $\lim_{t
ightarrow8^{-}}f(t)=\lim_{t
ightarrow8^{+}}f(t)$.

Step2: Calculate left - hand limit

$\lim_{t
ightarrow8^{-}}f(t)=\lim_{t
ightarrow8^{-}}(t^{2}-c)=8^{2}-c=64 - c$.

Step3: Calculate right - hand limit

$\lim_{t
ightarrow8^{+}}f(t)=\lim_{t
ightarrow8^{+}}(ct + 4)=8c+4$.

Step4: Set left - hand and right - hand limits equal

$64 - c=8c+4$.

Step5: Solve for c

Add $c$ to both sides: $64=9c + 4$. Subtract 4 from both sides: $60 = 9c$. Then $c=\frac{60}{9}=\frac{20}{3}$.

Answer:

$c=\frac{20}{3}$