Question

week 3 - sections 15 and 16 limits: problem (1 point) find the value of the constant ( c ) that makes the following function continuous on ( (-infty, infty) ). f(x) = \begin{cases} x^2 - c & \text{if } -infty < x < 9 \\ cx + 9 & \text{if } x geq 9 end{cases} ( c = \frac{36}{5} ) (with a checkmark)

Explanation:

Step1: Set equal at x=9

For continuity, $x^2 - c = cx + 9$ at $x=9$.

Step2: Substitute x=9

$9^2 - c = c(9) + 9$
$\implies 81 - c = 9c + 9$

Step3: Isolate c terms

$81 - 9 = 9c + c$
$\implies 72 = 10c$

Step4: Solve for c

$c = \frac{72}{10} = \frac{36}{5}$

Answer:

$\frac{36}{5}$