Given The Function $h(x) = 3 \sqrt{x}$, Which Statement Is True About $h(x)$?A. The Function Is Decreasing On The Interval $(-\infty, 0)$.B. The Function Is Increasing On The Interval \$(-\infty,

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Understanding the Properties of the Function h(x) = 3√x

In mathematics, functions are used to describe the relationship between variables. The given function h(x) = 3√x is a simple yet powerful example of a function that can be analyzed to understand its properties. In this article, we will explore the properties of the function h(x) = 3√x and determine which statement is true about it.

The function h(x) = 3√x is a square root function that has been multiplied by a constant factor of 3. This function is defined for all non-negative real numbers, i.e., x ≥ 0. The square root function is a fundamental function in mathematics that has many applications in various fields, including physics, engineering, and economics.

To determine whether the function h(x) = 3√x is increasing or decreasing, we need to analyze its derivative. The derivative of a function represents the rate of change of the function with respect to its input variable. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.

Derivative of h(x) = 3√x

To find the derivative of h(x) = 3√x, we can use the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1). In this case, we have h(x) = 3√x = 3x^(1/2). Applying the power rule, we get:

h'(x) = d(3x^(1/2))/dx = 3(1/2)x^((1/2)-1) = (3/2)x^(-1/2)

Analysis of the Derivative

Now that we have the derivative of h(x) = 3√x, we can analyze its behavior. The derivative h'(x) = (3/2)x^(-1/2) is a rational function that is defined for all non-negative real numbers, i.e., x ≥ 0. To determine whether the function is increasing or decreasing, we need to examine the sign of the derivative.

Sign of the Derivative

The derivative h'(x) = (3/2)x^(-1/2) is a rational function that has a positive numerator and a negative exponent. This means that the derivative is positive for all x > 0, and the function is increasing on the interval (0, ∞).

Based on the analysis of the derivative, we can conclude that the function h(x) = 3√x is increasing on the interval (0, ∞). This means that as x increases, h(x) also increases. Therefore, the correct statement about the function h(x) = 3√x is:

  • The function is increasing on the interval (0, ∞).

The correct answer is:

  • B. The function is increasing on the interval (0, ∞).

In conclusion, the function h(x) = 3√x is a simple yet powerful example of a function that can be analyzed to understand its properties. By examining the derivative of the function, we can determine whether the function is increasing or decreasing. In this case, we found that the function is increasing on the interval (0, ∞). This analysis has important implications for various fields, including physics, engineering, and economics, where the square root function is widely used.
Frequently Asked Questions (FAQs) about the Function h(x) = 3√x

In our previous article, we explored the properties of the function h(x) = 3√x and determined that it is increasing on the interval (0, ∞). In this article, we will answer some frequently asked questions (FAQs) about the function h(x) = 3√x.

Q: What is the domain of the function h(x) = 3√x?

A: The domain of the function h(x) = 3√x is all non-negative real numbers, i.e., x ≥ 0.

Q: What is the range of the function h(x) = 3√x?

A: The range of the function h(x) = 3√x is all non-negative real numbers, i.e., h(x) ≥ 0.

Q: Is the function h(x) = 3√x continuous?

A: Yes, the function h(x) = 3√x is continuous on its domain, i.e., x ≥ 0.

Q: Is the function h(x) = 3√x differentiable?

A: Yes, the function h(x) = 3√x is differentiable on its domain, i.e., x > 0.

Q: What is the derivative of the function h(x) = 3√x?

A: The derivative of the function h(x) = 3√x is h'(x) = (3/2)x^(-1/2).

Q: Is the function h(x) = 3√x increasing or decreasing?

A: The function h(x) = 3√x is increasing on the interval (0, ∞).

Q: What is the value of the function h(x) = 3√x at x = 0?

A: The value of the function h(x) = 3√x at x = 0 is h(0) = 0.

Q: What is the value of the function h(x) = 3√x at x = 1?

A: The value of the function h(x) = 3√x at x = 1 is h(1) = 3.

Q: Can the function h(x) = 3√x be inverted?

A: Yes, the function h(x) = 3√x can be inverted to obtain the inverse function h^(-1)(x) = (x/3)^2.

In conclusion, the function h(x) = 3√x is a simple yet powerful example of a function that can be analyzed to understand its properties. By answering some frequently asked questions (FAQs) about the function, we have gained a deeper understanding of its behavior and properties. We hope that this article has been helpful in clarifying any doubts about the function h(x) = 3√x.

For further reading and exploration, we recommend the following resources:

  • Calculus textbooks: For a comprehensive understanding of calculus and its applications, we recommend consulting calculus textbooks such as "Calculus" by Michael Spivak or "Calculus: Early Transcendentals" by James Stewart.
  • Online resources: For online resources and tutorials on calculus and its applications, we recommend visiting websites such as Khan Academy, MIT OpenCourseWare, or Wolfram Alpha.
  • Mathematical software: For mathematical software and tools, we recommend using software such as Mathematica, Maple, or MATLAB.

In conclusion, the function h(x) = 3√x is a fundamental example of a function that can be analyzed to understand its properties. By answering some frequently asked questions (FAQs) about the function, we have gained a deeper understanding of its behavior and properties. We hope that this article has been helpful in clarifying any doubts about the function h(x) = 3√x.