Chapter 9: Q77. (page 535)

What is the range of the function $f (x) = - 4 x^{2} - \frac{1}{2} ?$
$A \{all integers less than or equal to \frac{1}{2}\} B {all nonnegative integers} C {all real numbers} D {all real numbers less than or equal to - \frac{}{}}$

Short Answer

Expert verified

The range of the given function is all real numbers less than or equal to $- \frac{1}{2}$ .

Step by step solution

Step 1. Identify.

The function $f (x) = - 4 x^{2} - \frac{1}{2}$ is a parabola with $a = - 4, b = 0$ and $c = - \frac{1}{2}$ . As a is negative, the graph opens downward, and the function has a maximum value. The range is all real numbers less than or equal to the maximum value.

Step 2. Find x coordinate of vertex.

X coordinate of vertex is given by $x = \frac{- b}{2 a}$ . Substitute $- 4$ for a, 0 for b and $- \frac{1}{2}$ for c

$x = \frac{- 0}{2 (- 4)} = 0$

Step 3. Find the value of y.

Substitute x-coordinate of vertex into the original equation,

$f (x) = - 4 x^{2} - \frac{1}{2} = - 4 {(0)}^{2} - \frac{1}{2} = - \frac{1}{2}$

Therefore, the vertex is at $(0, - \frac{1}{2})$ .

The maximum value is $- \frac{1}{2}$ . The range is all real numbers less than or equal to the maximum value, or all real numbers less than or equal to $- \frac{1}{2}$ . Option D is the required answer.

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What is the range of the function $f (x) = - 4 x^{2} - \frac{1}{2} ?$
$A \{all integers less than or equal to \frac{1}{2}\} B {all nonnegative integers} C {all real numbers} D {all real numbers less than or equal to - \frac{}{}}$

Short Answer

Step by step solution

Step 1. Identify.

Step 2. Find x coordinate of vertex.

Step 3. Find the value of y.

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What is the range of the function fx=−4x2−12?Aallintegerslessthanorequalto12B{allnonnegativeintegers}C{allrealnumbers}D{allrealnumberslessthanorequalto−}

Short Answer

Step by step solution

Step 1. Identify.

Step 2. Find x coordinate of vertex.

Step 3. Find the value of y.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Theoretical and Mathematical Physics

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Calculus

Study anywhere. Anytime. Across all devices.

What is the range of the function $f (x) = - 4 x^{2} - \frac{1}{2} ?$
$A \{all integers less than or equal to \frac{1}{2}\} B {all nonnegative integers} C {all real numbers} D {all real numbers less than or equal to - \frac{}{}}$