Question

which graph represents the function $f(x) = \frac{1}{x + 3} - 2$?

Explanation:

To determine the graph of \( f(x) = \frac{1}{x + 3} - 2 \), we analyze its key features:

Step 1: Identify the Vertical Asymptote

The vertical asymptote of a rational function \( \frac{1}{x - h} + k \) occurs where the denominator is zero (\( x - h = 0 \)). For \( f(x) = \frac{1}{x + 3} - 2 \) (rewritten as \( \frac{1}{x - (-3)} + (-2) \)), the denominator \( x + 3 = 0 \) when \( x = -3 \). Thus, the vertical asymptote is \( x = -3 \).

Step 2: Identify the Horizontal Asymptote

For a rational function of the form \( \frac{1}{x - h} + k \), the horizontal asymptote is \( y = k \) (since the degree of the numerator is less than the degree of the denominator). Here, \( k = -2 \), so the horizontal asymptote is \( y = -2 \).

Step 3: Analyze the Shape and Behavior

• For \( x > -3 \) (right of the vertical asymptote), as \( x \to -3^+ \), \( x + 3 \to 0^+ \), so \( \frac{1}{x + 3} \to +\infty \), and \( f(x) \to +\infty - 2 = +\infty \). As \( x \to +\infty \), \( \frac{1}{x + 3} \to 0 \), so \( f(x) \to 0 - 2 = -2 \) (approaching the horizontal asymptote from above).
• For \( x < -3 \) (left of the vertical asymptote), as \( x \to -3^- \), \( x + 3 \to 0^- \), so \( \frac{1}{x + 3} \to -\infty \), and \( f(x) \to -\infty - 2 = -\infty \). As \( x \to -\infty \), \( \frac{1}{x + 3} \to 0 \), so \( f(x) \to 0 - 2 = -2 \) (approaching the horizontal asymptote from below).

Step 4: Compare with Graphs

The graph of \( f(x) = \frac{1}{x + 3} - 2 \) will have:

• A vertical asymptote at \( x = -3 \).
• A horizontal asymptote at \( y = -2 \).
• Two branches: one in the region \( x > -3 \) (approaching \( y = -2 \) from above as \( x \to +\infty \)) and one in \( x < -3 \) (approaching \( y = -2 \) from below as \( x \to -\infty \)).

To confirm, match these features (asymptotes, end - behavior, and shape) with the given graph options. The correct graph will align with these key characteristics.

(Note: Since the specific graph options are not fully visible, use the above analysis to identify the graph with vertical asymptote \( x = -3 \), horizontal asymptote \( y = -2 \), and the described branch behavior.)

Answer:

To determine the graph of \( f(x) = \frac{1}{x + 3} - 2 \), we analyze its key features:

Step 1: Identify the Vertical Asymptote

The vertical asymptote of a rational function \( \frac{1}{x - h} + k \) occurs where the denominator is zero (\( x - h = 0 \)). For \( f(x) = \frac{1}{x + 3} - 2 \) (rewritten as \( \frac{1}{x - (-3)} + (-2) \)), the denominator \( x + 3 = 0 \) when \( x = -3 \). Thus, the vertical asymptote is \( x = -3 \).

Step 2: Identify the Horizontal Asymptote

For a rational function of the form \( \frac{1}{x - h} + k \), the horizontal asymptote is \( y = k \) (since the degree of the numerator is less than the degree of the denominator). Here, \( k = -2 \), so the horizontal asymptote is \( y = -2 \).

Step 3: Analyze the Shape and Behavior

• For \( x > -3 \) (right of the vertical asymptote), as \( x \to -3^+ \), \( x + 3 \to 0^+ \), so \( \frac{1}{x + 3} \to +\infty \), and \( f(x) \to +\infty - 2 = +\infty \). As \( x \to +\infty \), \( \frac{1}{x + 3} \to 0 \), so \( f(x) \to 0 - 2 = -2 \) (approaching the horizontal asymptote from above).
• For \( x < -3 \) (left of the vertical asymptote), as \( x \to -3^- \), \( x + 3 \to 0^- \), so \( \frac{1}{x + 3} \to -\infty \), and \( f(x) \to -\infty - 2 = -\infty \). As \( x \to -\infty \), \( \frac{1}{x + 3} \to 0 \), so \( f(x) \to 0 - 2 = -2 \) (approaching the horizontal asymptote from below).

Step 4: Compare with Graphs

The graph of \( f(x) = \frac{1}{x + 3} - 2 \) will have:

• A vertical asymptote at \( x = -3 \).
• A horizontal asymptote at \( y = -2 \).
• Two branches: one in the region \( x > -3 \) (approaching \( y = -2 \) from above as \( x \to +\infty \)) and one in \( x < -3 \) (approaching \( y = -2 \) from below as \( x \to -\infty \)).

To confirm, match these features (asymptotes, end - behavior, and shape) with the given graph options. The correct graph will align with these key characteristics.

(Note: Since the specific graph options are not fully visible, use the above analysis to identify the graph with vertical asymptote \( x = -3 \), horizontal asymptote \( y = -2 \), and the described branch behavior.)