Questions: Sketch a possible graph of a function that satisfies f(-1)=-7; lim x to (-1)^- f(x)=-2; lim x to (-1)^+ f(x)=-7

Transcript text: Sketch a possible graph of a function that satisfies \[ f(-1)=-7 ; \lim _{x \to (-1)^{-}} f(x)=-2 ; \lim _{x \to (-1)^{+}} f(x)=-7 \]

Solution

Solution Steps

Step 1: Understand the Given Conditions

We are given the following conditions for the function \( f(x) \):

\( f(-1) = -7 \)
\( \lim_{{x \to -1}^-} f(x) = -2 \)
\( \lim_{{x \to -1}^+} f(x) = -7 \)

Step 2: Interpret the Conditions

\( f(-1) = -7 \) means the function value at \( x = -1 \) is \( -7 \).
\( \lim_{{x \to -1}^-} f(x) = -2 \) means as \( x \) approaches \( -1 \) from the left, the function approaches \( -2 \).
\( \lim_{{x \to -1}^+} f(x) = -7 \) means as \( x \) approaches \( -1 \) from the right, the function approaches \( -7 \).

Step 3: Sketch the Graph

Plot the point \( (-1, -7) \) on the graph.
Draw a left-hand limit approaching \( -2 \) as \( x \) approaches \( -1 \) from the left. This means the graph should approach the point \( (-1, -2) \) but not touch it.
Draw a right-hand limit approaching \( -7 \) as \( x \) approaches \( -1 \) from the right. This means the graph should approach the point \( (-1, -7) \) and touch it since \( f(-1) = -7 \).