Given The Functions: ${ F(x) = 4x, \quad G(x) = |x + 2|, \quad H(x) = \frac{1}{x - 6} }$Evaluate The Function { \left(\frac{h}{f}\right)(x)$}$ For { X = 0$}$. Write The Answer As An Integer Or A Simplified

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Introduction

In mathematics, functions are used to describe relationships between variables. Given two functions, f(x) and g(x), we can perform various operations on them, such as addition, subtraction, multiplication, and division. However, when dividing two functions, we need to ensure that the denominator is not equal to zero. In this article, we will evaluate the function (h/f)(x) for x = 0, where f(x) = 4x, g(x) = |x + 2|, and h(x) = 1/(x - 6).

Understanding the Functions

Before we proceed with the evaluation, let's understand the given functions.

Function f(x) = 4x

The function f(x) = 4x is a linear function that takes an input x and multiplies it by 4. This function is defined for all real numbers x.

Function g(x) = |x + 2|

The function g(x) = |x + 2| is an absolute value function that takes an input x and returns the absolute value of x + 2. This function is also defined for all real numbers x.

Function h(x) = 1/(x - 6)

The function h(x) = 1/(x - 6) is a rational function that takes an input x and returns the reciprocal of x - 6. This function is defined for all real numbers x except x = 6, where the denominator is equal to zero.

Evaluating the Function (h/f)(x)

To evaluate the function (h/f)(x), we need to divide the function h(x) by the function f(x). This can be written as:

(h/f)(x) = h(x) / f(x)

Substituting the given functions, we get:

(h/f)(x) = (1/(x - 6)) / (4x)

To simplify this expression, we can multiply the numerator and denominator by 4x, which gives us:

(h/f)(x) = (4x) / ((x - 6)(4x))

Simplifying further, we get:

(h/f)(x) = 1 / (x - 6)

Evaluating the Function (h/f)(x) for x = 0

Now that we have simplified the function (h/f)(x), we can evaluate it for x = 0.

(h/f)(0) = 1 / (0 - 6)

(h/f)(0) = 1 / (-6)

(h/f)(0) = -1/6

Therefore, the value of the function (h/f)(x) for x = 0 is -1/6.

Conclusion

In this article, we evaluated the function (h/f)(x) for x = 0, where f(x) = 4x, g(x) = |x + 2|, and h(x) = 1/(x - 6). We simplified the function (h/f)(x) and found that its value for x = 0 is -1/6. This demonstrates the importance of understanding and simplifying functions in mathematics.

Frequently Asked Questions

  • What is the value of the function (h/f)(x) for x = 0?
  • How do we simplify the function (h/f)(x)?
  • What are the given functions f(x), g(x), and h(x)?

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Algebra, 2nd edition, Michael Artin
  • [3] Functions, 1st edition, David Guichard

Further Reading

  • [1] Evaluating Functions, Khan Academy
  • [2] Simplifying Functions, Mathway
  • [3] Rational Functions, Wolfram MathWorld

Introduction

In our previous article, we evaluated the function (h/f)(x) for x = 0, where f(x) = 4x, g(x) = |x + 2|, and h(x) = 1/(x - 6). We simplified the function (h/f)(x) and found that its value for x = 0 is -1/6. In this article, we will answer some frequently asked questions related to the evaluation of the function (h/f)(x) for x = 0.

Q&A

Q: What is the value of the function (h/f)(x) for x = 0?

A: The value of the function (h/f)(x) for x = 0 is -1/6.

Q: How do we simplify the function (h/f)(x)?

A: To simplify the function (h/f)(x), we can multiply the numerator and denominator by 4x, which gives us:

(h/f)(x) = (4x) / ((x - 6)(4x))

Simplifying further, we get:

(h/f)(x) = 1 / (x - 6)

Q: What are the given functions f(x), g(x), and h(x)?

A: The given functions are:

  • f(x) = 4x
  • g(x) = |x + 2|
  • h(x) = 1/(x - 6)

Q: Why is the function h(x) = 1/(x - 6) not defined for x = 6?

A: The function h(x) = 1/(x - 6) is not defined for x = 6 because the denominator is equal to zero, which would result in division by zero.

Q: Can we evaluate the function (h/f)(x) for any value of x?

A: Yes, we can evaluate the function (h/f)(x) for any value of x, except x = 6, where the denominator of the function h(x) is equal to zero.

Q: How do we handle the absolute value function g(x) = |x + 2|?

A: The absolute value function g(x) = |x + 2| is not used in the evaluation of the function (h/f)(x) for x = 0. However, it is an important function in mathematics and is used in various applications.

Q: Can we simplify the function (h/f)(x) further?

A: Yes, we can simplify the function (h/f)(x) further by canceling out the common factors in the numerator and denominator. However, in this case, the function (h/f)(x) is already simplified.

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the function (h/f)(x) for x = 0. We provided detailed explanations and examples to help clarify the concepts. We hope that this article has been helpful in understanding the evaluation of the function (h/f)(x) for x = 0.

Frequently Asked Questions

  • What is the value of the function (h/f)(x) for x = 0?
  • How do we simplify the function (h/f)(x)?
  • What are the given functions f(x), g(x), and h(x)?
  • Why is the function h(x) = 1/(x - 6) not defined for x = 6?
  • Can we evaluate the function (h/f)(x) for any value of x?
  • How do we handle the absolute value function g(x) = |x + 2|?
  • Can we simplify the function (h/f)(x) further?

References

  • [1] Calculus, 3rd edition, Michael Spivak
  • [2] Algebra, 2nd edition, Michael Artin
  • [3] Functions, 1st edition, David Guichard

Further Reading

  • [1] Evaluating Functions, Khan Academy
  • [2] Simplifying Functions, Mathway
  • [3] Rational Functions, Wolfram MathWorld