Write The Function In Standard Form:${ G(x) = \frac{2x + 1}{x - 2} }$

A rational function is a function that can be expressed as the ratio of two polynomials. The standard form of a rational function is a way of expressing the function in a simplified and compact form. In this article, we will discuss how to write a rational function in standard form.

What is Standard Form?

The standard form of a rational function is a way of expressing the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

where a, b, c, and d are constants, and x is the variable.

Why is Standard Form Important?

Standard form is important because it allows us to simplify and manipulate rational functions more easily. It also makes it easier to identify the domain and range of the function.

How to Write a Rational Function in Standard Form

To write a rational function in standard form, we need to follow these steps:

1. Factor the numerator and denominator: Factor the numerator and denominator of the rational function, if possible.
2. Cancel out any common factors: Cancel out any common factors between the numerator and denominator.
3. Write the function in the form: Write the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

Example 1: Writing a Rational Function in Standard Form

Let's consider the rational function:

${ g(x) = \frac{2x + 1}{x - 2} }$

To write this function in standard form, we need to factor the numerator and denominator, and then cancel out any common factors.

Step 1: Factor the numerator and denominator

The numerator 2x + 1 can be factored as:

${ 2x + 1 = (2x + 1) }$

The denominator x - 2 can be factored as:

${ x - 2 = (x - 2) }$

Step 2: Cancel out any common factors

There are no common factors between the numerator and denominator, so we cannot cancel out any factors.

Step 3: Write the function in the form

The function can be written in the form:

${ g(x) = \frac{2x + 1}{x - 2} }$

This is already in standard form, so we are done.

Example 2: Writing a Rational Function in Standard Form

Let's consider the rational function:

${ g(x) = \frac{x^2 + 4x + 4}{x^2 + 5x + 6} }$

To write this function in standard form, we need to factor the numerator and denominator, and then cancel out any common factors.

Step 1: Factor the numerator and denominator

The numerator x^2 + 4x + 4 can be factored as:

${ x^2 + 4x + 4 = (x + 2)^2 }$

The denominator x^2 + 5x + 6 can be factored as:

${ x^2 + 5x + 6 = (x + 3)(x + 2) }$

Step 2: Cancel out any common factors

We can cancel out the common factor (x + 2) between the numerator and denominator.

Step 3: Write the function in the form

The function can be written in the form:

${ g(x) = \frac{x + 2}{x + 3} }$

This is in standard form, so we are done.

Conclusion

In this article, we discussed how to write a rational function in standard form. We also provided two examples of how to write a rational function in standard form. Standard form is an important concept in mathematics, and it allows us to simplify and manipulate rational functions more easily.

Key Takeaways

• The standard form of a rational function is a way of expressing the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

• To write a rational function in standard form, we need to factor the numerator and denominator, and then cancel out any common factors.
• Standard form is important because it allows us to simplify and manipulate rational functions more easily.
• It also makes it easier to identify the domain and range of the function.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Standard Form of a Rational Function" by Purplemath

Further Reading

• "Rational Functions" by Khan Academy
• "Standard Form of a Rational Function" by Mathway

In our previous article, we discussed how to write a rational function in standard form. In this article, we will answer some frequently asked questions about the standard form of a rational function.

Q: What is the standard form of a rational function?

A: The standard form of a rational function is a way of expressing the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

where a, b, c, and d are constants, and x is the variable.

Q: Why is standard form important?

A: Standard form is important because it allows us to simplify and manipulate rational functions more easily. It also makes it easier to identify the domain and range of the function.

Q: How do I write a rational function in standard form?

A: To write a rational function in standard form, you need to follow these steps:

1. Factor the numerator and denominator: Factor the numerator and denominator of the rational function, if possible.
2. Cancel out any common factors: Cancel out any common factors between the numerator and denominator.
3. Write the function in the form: Write the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

Q: What if I have a rational function with a quadratic numerator and denominator?

A: If you have a rational function with a quadratic numerator and denominator, you can factor the numerator and denominator, and then cancel out any common factors. For example:

${ g(x) = \frac{x^2 + 4x + 4}{x^2 + 5x + 6} }$

You can factor the numerator and denominator as:

${ g(x) = \frac{(x + 2)^2}{(x + 3)(x + 2)} }$

Then, you can cancel out the common factor (x + 2) between the numerator and denominator.

Q: What if I have a rational function with a cubic numerator and denominator?

A: If you have a rational function with a cubic numerator and denominator, you can factor the numerator and denominator, and then cancel out any common factors. For example:

${ g(x) = \frac{x^3 + 6x^2 + 11x + 6}{x^3 + 8x^2 + 19x + 12} }$

You can factor the numerator and denominator as:

${ g(x) = \frac{(x + 1)(x + 2)(x + 3)}{(x + 1)(x + 2)(x + 3)} }$

Then, you can cancel out the common factors (x + 1), (x + 2), and (x + 3) between the numerator and denominator.

Q: Can I have a rational function with a non-integer coefficient?

A: Yes, you can have a rational function with a non-integer coefficient. For example:

${ g(x) = \frac{2x + 1}{x - 2} }$

This rational function has a non-integer coefficient of 2 in the numerator.

Q: Can I have a rational function with a variable coefficient?

A: Yes, you can have a rational function with a variable coefficient. For example:

${ g(x) = \frac{ax + b}{cx + d} }$

This rational function has a variable coefficient a in the numerator.

Conclusion

In this article, we answered some frequently asked questions about the standard form of a rational function. We hope that this article has been helpful in clarifying any doubts you may have had about the standard form of a rational function.

Key Takeaways

• The standard form of a rational function is a way of expressing the function in the form:

${ g(x) = \frac{ax + b}{cx + d} }$

• To write a rational function in standard form, you need to factor the numerator and denominator, and then cancel out any common factors.
• Standard form is important because it allows us to simplify and manipulate rational functions more easily.
• It also makes it easier to identify the domain and range of the function.

References

• [1] "Rational Functions" by Math Open Reference
• [2] "Standard Form of a Rational Function" by Purplemath

Further Reading

• "Rational Functions" by Khan Academy
• "Standard Form of a Rational Function" by Mathway