Select the correct answer.
What is the zero of function [tex]$f$[/tex]?
[tex]$f(x)=7 \sqrt[3]{x+12}-21$[/tex]
A. [tex]$x=0$[/tex]
B. [tex]$x=15$[/tex]
C. [tex]$x=-5$[/tex]
D. [tex]$x=-12$[/tex]
Answer (2)
Set the function equal to zero: 7 3 x + 12 − 21 = 0 .
Isolate the cube root: 3 x + 12 = 3 .
Cube both sides: x + 12 = 27 .
Solve for x : x = 15 , so the zero of the function is 15 .
Explanation
Understanding the Problem We are given the function f ( x ) = 7 3 x + 12 − 21 and asked to find its zero. This means we need to find the value of x for which f ( x ) = 0 .
Setting up the Equation To find the zero of the function, we set f ( x ) = 0 and solve for x : 7 3 x + 12 − 21 = 0
Isolating the Cube Root Term First, we isolate the term with the cube root by adding 21 to both sides of the equation: 7 3 x + 12 = 21
Dividing by 7 Next, we divide both sides by 7: 3 x + 12 = 3
Cubing Both Sides To eliminate the cube root, we cube both sides of the equation: ( 3 x + 12 ) 3 = 3 3 x + 12 = 27
Solving for x Now, we solve for x by subtracting 12 from both sides: x = 27 − 12 x = 15
Final Answer Therefore, the zero of the function f ( x ) is x = 15 .
Examples
Understanding zeros of functions is crucial in many real-world applications. For instance, in physics, finding the zero of a function representing the height of a projectile helps determine when the projectile hits the ground. In economics, zeros of cost or profit functions can indicate break-even points. In engineering, zeros can represent equilibrium states or critical points in system behavior. Thus, mastering the concept of finding zeros is essential for problem-solving across various disciplines.
To find the zero of the function f ( x ) = 7 3 x + 12 − 21 , we set it equal to zero and solve for x . By isolating the cube root term and cubing both sides, we find that the zero is x = 15 . Thus, the correct choice is B. x = 15 .
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