Questions: Graph the function. F(x) = 3-x, if x ≤ 2 1+2x, if x > 2

Solution Steps

The given function is a piecewise function defined as: \[ F(x)=\left\{\begin{array}{ll} 3-x, & x \leq 2 \\ 1+2x, & x>2 \end{array}\right. \]

To understand the behavior of the function, we can evaluate it at specific points:

• For \( x \leq 2 \):
  • \( F(0) = 3 - 0 = 3 \)
  • \( F(1) = 3 - 1 = 2 \)
  • \( F(2) = 3 - 2 = 1 \)
• For \( x > 2 \):
  • \( F(3) = 1 + 2 \cdot 3 = 7 \)
  • \( F(4) = 1 + 2 \cdot 4 = 9 \)

To check the continuity at \( x = 2 \):

• Left-hand limit: \( \lim_{{x \to 2^-}} F(x) = 3 - 2 = 1 \)
• Right-hand limit: \( \lim_{{x \to 2^+}} F(x) = 1 + 2 \cdot 2 = 5 \)

Since the left-hand limit and right-hand limit are not equal, the function is not continuous at \( x = 2 \).

Final Answer