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The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function?

all real numbers less than or equal to 4
all real numbers less than or equal to −3
all real numbers greater than or equal to 4
all real numbers greater than or equal to −3

User Phildo
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2 Answers

5 votes
The function f(x) = −(x + 5)(x + 1) is shown.What is the range of the function?

all real numbers less than or equal to 4


User Csnewb
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9.2k points
7 votes

we have


f(x)=-(x+5)(x+1)

we know that

The equation of a vertical parabola into vertex form is equal to


y=a(x-h)^(2)+k

where

(h,k) is the vertex of the parabola

if
a>0 ----> the parabola open upward (vertex is a minimum)

if
a<0 ----> the parabola open downward (vertex is a maximum)

In this problem convert the quadratic equation into vertex form

so


f(x)=-(x+5)(x+1)\\f(x)=-(x^(2) +x+5x+5)\\f(x)=-( x^(2) +6x+5)


f(x)=-x^(2)-6x-5

Group terms that contain the same variable, and move the constant to the opposite side of the equation


f(x)+5=-(x^(2)+6x)

Complete the square. Remember to balance the equation by adding the same constants to each side


f(x)+5-9=-(x^(2)+6x+9)


f(x)-4=-(x^(2)+6x+9)

Rewrite as perfect squares


f(x)-4=-(x+3)^(2)


f(x)=-(x+3)^(2)+4

This is a vertical parabola open down (vertex is a maximum)

the vertex is the point
(-3,4)

The range is the interval--------> (-∞,4]


y\leq 4

All real numbers less than or equal to
4

therefore

the answer is the option

All real numbers less than or equal to
4

User Shaydawg
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7.9k points

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