Question

the graph of the piecewise function f(x) is shown. what is the range of f(x)? {x|-2≤x<4} {x|-2<x≤4} {y|-5<y<-1} {y|-5≤y≤-1}

Explanation:

Step1: Recall Range Definition

Range of a function is the set of all possible \( y \)-values (outputs) the function can take. So we analyze the \( y \)-coordinates of the graph.

Step2: Analyze the Graph's \( y \)-Values

• The leftmost point (closed dot) has a \( y \)-value of \( -5 \) (since it's a closed dot, \( y = -5 \) is included).
• The peak (closed dot at \( x = 2 \)) has a \( y \)-value of \( -1 \) (closed dot, so \( y = -1 \) is included).
• The rightmost point (open dot) has a \( y \)-value of \( -2 \), but we look at the minimum and maximum \( y \)-values covered. The graph spans from \( y = -5 \) (inclusive) up to \( y = -1 \) (inclusive), with all \( y \)-values between them.

Step3: Evaluate the Options

• First two options are about \( x \)-values (domain), so eliminate them.
• Third option: \( \{y \mid -5 < y < -1\} \) excludes \( -5 \) and \( -1 \), but our graph has closed dots at \( y = -5 \) and \( y = -1 \), so they should be included.
• Fourth option: \( \{y \mid -5 \leq y \leq -1\} \) includes \( -5 \) (closed dot) and \( -1 \) (closed dot), and all \( y \)-values between, which matches.

Answer:

\(\{y \mid -5 \leq y \leq -1\}\) (the fourth option)