How To Find The Domain of a Function - Radicals, Fractions & Square Roots - Interval Notation

Summary

This script offers a comprehensive guide on determining the domain of various mathematical functions. It explains that linear and polynomial functions have a domain of all real numbers, while rational functions require the exclusion of values that would result in a zero denominator. The script also covers how to find the domain for square root functions, emphasizing that the radicand must be non-negative. It demonstrates the process with examples, including rational functions with square roots in both the numerator and denominator, and illustrates the use of interval notation to represent the domain.

Takeaways

• 📚 The domain of a linear function, like 2x - 7, is all real numbers, ranging from negative infinity to positive infinity.
• 📐 For quadratic functions and polynomials, the domain remains all real numbers, as there are no restrictions on the variable x.
• 🚫 In rational functions, the domain excludes values that make the denominator zero, as these would create vertical asymptotes.
• 📉 To represent the domain of a rational function, use interval notation with open circles at excluded points and lines extending to infinity.
• 🔍 For rational functions, factor the denominator and set each factor not equal to zero to find the domain.
• 🛑 If a function includes a square root, the domain excludes negative values inside the square root, as even roots of negative numbers are not real.
• 📌 When a square root is in the denominator, the inside must be strictly greater than zero to avoid division by zero.
• 🔄 For functions with square roots in both the numerator and denominator, find the intersection of the domains from each part.
• 📈 When determining the domain of a square root function, set the expression inside the root to be greater than or equal to zero.
• 📊 Use a number line and test points to determine which intervals are valid for the domain when dealing with complex expressions involving square roots.
• 📝 The domain of a function can be found by considering restrictions from denominators, square roots, and other function-specific limitations.

Q & A

What is the domain of the function 2x - 7?

The domain of the function 2x - 7 is all real numbers, as it is a linear function without any restrictions on the input values.

How does the domain of a quadratic function compare to that of a linear function?

The domain of a quadratic function, like any polynomial function, is also all real numbers, since there are no restrictions on the input values that would make the function undefined.

What is the main consideration when finding the domain of a rational function?

For rational functions, the main consideration is to ensure that the denominator is never zero, as this would make the function undefined.

How do you represent the domain of a function with restrictions using interval notation?

In interval notation, you use parentheses to indicate that an endpoint is not included, and brackets to indicate that an endpoint is included. For example, (2, ∞) represents all numbers greater than 2 but not including 2.

What is the domain of the function 5/(x - 2)?

The domain of the function 5/(x - 2) is all real numbers except x = 2, as the denominator cannot be zero. In interval notation, this is represented as (-∞, 2) U (2, ∞).

How do you determine the domain of a function with a square root in the denominator?

For a function with a square root in the denominator, you set the expression inside the square root greater than zero, as you cannot have zero in the denominator. The domain excludes any values that would make the square root expression equal to zero.

What is the domain of the square root function √(x - 4)?

The domain of the square root function √(x - 4) is x ≥ 4, as the expression inside the square root must be greater than or equal to zero.

How do you find the domain of a function with a square root in both the numerator and the denominator?

For a function with square roots in both the numerator and the denominator, you find the intersection of the domains determined by setting the expressions inside each square root to be greater than zero for the numerator and strictly greater than zero for the denominator.

What is the domain of the function (2x - 3) / (x^2 + 4)?

The domain of the function (2x - 3) / (x^2 + 4) is all real numbers, as x^2 will never be equal to -4, and thus the denominator is never zero.

How do you determine if a region is included in the domain when dealing with square roots?

When dealing with square roots, you determine if a region is included in the domain by testing values within that region to see if they result in a positive value under the square root, as negative values are not allowed.

What is the domain of a function with a square root in the numerator and a quadratic expression in the denominator?

The domain of such a function is determined by setting the expression in the denominator to be greater than zero and ensuring the expression in the numerator is greater than or equal to zero. The domain excludes any values that would make the denominator zero.