Questions: Find the value of c such that the function is continuous on the entire real number line. f(x) = x+3, x ≤ 4 c x+9, x > 4 c=

Transcript text: Find the value of $c$ such that the function is continuous on the entire real number line. \[ f(x)=\left\{\begin{array}{ll} x+3, & x \leq 4 \\ c x+9, & x>4 \end{array}\right. \] $c=$

Solution

Solution Steps

Step 1: Identify the Point of Discontinuity

The point of discontinuity, $x_0$, is at $x = 4$.

Step 2: Set Up the Equation for Continuity

To ensure continuity at $x_0$, we equate the limits of $g(x)$ and $h(x)$ at this point: \[\lim_{x o 4^-} g(x) = \lim_{x o 4^+} h(x)\] \[\lim_{x o 4^-} x+3 = \lim_{x o 4^+} c*x+9\]

Step 3: Solve the Equation for $c$

Solving the equation, we find that $c$ must satisfy: 7 = 4 c + 9 This yields a solution for $c$: -0.5