Question

what is the range of the function $f(x) = -(x + 3)^2 + 7$?
all real numbers less than or equal to 7
all real numbers less than or equal to $-3$
all real numbers greater than or equal to $-3$
all real numbers greater than or equal to 7

Explanation:

Step1: Analyze squared term behavior

For any real $x$, $(x+3)^2 \geq 0$

Step2: Apply negative coefficient

Multiply by $-1$: $-(x+3)^2 \leq 0$

Step3: Add constant to find range

Add 7: $-(x+3)^2 + 7 \leq 7$

Answer:

all real numbers less than or equal to 7