Which statement is true of the function $f(x)=-\sqrt[3]{x}$? Select three options.
The function is always increasing.
The function has a domain of all real numbers.
The function has a range of ${y \mid-\infty
The function is a reflection of $y=\sqrt[3]{x}$.
The function passes through the point $(3,-27)$.
Answer (1)
The function f ( x ) = − 3 x has a domain of all real numbers.
The function f ( x ) = − 3 x has a range of all real numbers, represented as y ∣ − ∞ < y < ∞ .
The function f ( x ) = − 3 x is a reflection of y = 3 x across the x-axis.
The three true statements are identified, and the final answer is: The function has a domain of all real numbers, the function has a range of y ∣ − ∞ < y < ∞ , and the function is a reflection of y = 3 x .
The function has a domain of all real numbers, the function has a range of y ∣ − ∞ < y < ∞ , and the function is a reflection of y = 3 x
Explanation
Analyzing the problem We are given the function f ( x ) = − 3 x and asked to determine which three statements are true. Let's analyze each statement.
Checking if the function is increasing
The function is always increasing. To determine if the function is increasing or decreasing, we can consider two values x 1 and x 2 such that x 1 < x 2 . If f(x_2)"> f ( x 1 ) > f ( x 2 ) , the function is decreasing, and if f ( x 1 ) < f ( x 2 ) , the function is increasing. Let's take x 1 = 1 and x 2 = 2 . Then f ( 1 ) = − 3 1 = − 1 and f ( 2 ) = − 3 2 ≈ − 1.26 . Since -1.26"> − 1 > − 1.26 , the function is decreasing. Therefore, the statement "The function is always increasing" is false.
Checking the domain
The function has a domain of all real numbers. Since we are taking the cube root of x , we can input any real number for x . For example, 3 − 8 = − 2 . Therefore, the domain of the function is all real numbers. The statement "The function has a domain of all real numbers" is true.
Checking the range
The function has a range of y ∣ − ∞ < y < ∞ .
Since we can input any real number for x , the cube root of x can also be any real number. Multiplying by − 1 doesn't change this. Therefore, the range of the function is all real numbers. The statement "The function has a range of y ∣ − ∞ < y < ∞ " is true.
Checking for reflection
The function is a reflection of y = 3 x .
The function f ( x ) = − 3 x is the reflection of y = 3 x across the x-axis. Therefore, the statement "The function is a reflection of y = 3 x " is true.
Checking the point
The function passes through the point ( 3 , − 27 ) .
To check if the function passes through the point ( 3 , − 27 ) , we substitute x = 3 into the function: f ( 3 ) = − 3 3 ≈ − 1.44 . Since f ( 3 ) = − 27 , the function does not pass through the point ( 3 , − 27 ) . Therefore, the statement "The function passes through the point ( 3 , − 27 ) " is false.
Final Answer Therefore, the three true statements are:
The function has a domain of all real numbers.
The function has a range of y ∣ − ∞ < y < ∞ .
The function is a reflection of y = 3 x .
Examples
Understanding the properties of functions, such as domain, range, and whether they are increasing or decreasing, is crucial in many real-world applications. For example, in physics, the velocity of an object under constant deceleration can be modeled by a decreasing function. Knowing the domain and range of this function helps determine the time interval over which the model is valid and the possible range of velocities the object can have. Similarly, in economics, understanding the domain and range of a cost function can help a company determine the range of production levels for which the cost model is applicable and the possible range of costs the company might incur.
