Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the domain of the function. $$g(y)=\frac{y+2}{\sqrt{y-10}}$$

Step-by-Step Solution

Answer

The domain of the function \(g(y)=\frac{y+2}{\sqrt{y-10}}\) is \(y > 10\).

Step 1: Analyze the denominator

Identify that the function will be undefined whenever the denominator is zero or negative. Therefore, set the expression under the square root greater than zero: \(y - 10 > 0\).

Step 2: Solve for y

While solving \(y-10>0\) it gives y>10.

Step 3: Conclusion

The function \(g(y)=\frac{y+2}{\sqrt{y-10}}\) will be defined and real for all values y such that \(y > 10\). Therefore, the domain of the function is \(y > 10\).

Key Concepts

Rational functions

Rational functions are mathematical expressions formed by the ratio of two polynomials. In simpler terms, you can think of them as fractions, where both the numerator and the denominator are polynomials. For instance, with the function \( g(y)=\frac{y+2}{\sqrt{y-10}} \), we see that it is a rational function with polynomials and incorporates a square root in the denominator.
To determine when this rational function is defined, consider:

• The denominator cannot be zero. If it becomes zero, the function is undefined due to division by zero.
• For a well-defined rational function, the denominator must be greater than zero in case of a square root, ensuring all operations are valid and result in real numbers.

Understanding rational functions is fundamental because they are often encountered in fields like physics and engineering, where they model various phenomena. By examining each part, specifically the denominator, we identify values that do or do not belong in the function's domain.

Square root functions

Square root functions involve the square root of a variable or expression. They are interesting because they only provide real numbers for inputs under specific conditions. The function mentioned, \(g(y)=\frac{y+2}{\sqrt{y-10}} \), has a square root in the denominator. This affects its domain significantly.

• Square root expressions require that what's inside the root is not negative. This means that for real numbers, the expression under the square root needs to be greater than zero.
• In our exercise, the expression \( \sqrt{y - 10} \) demands \( y - 10 \) to be greater than zero. Therefore, \( y > 10 \).

By understanding these conditions, you can determine when a square root function is valid and defined, which is key for calculating real-world problems accurately. Square root functions often appear in geometry and science, particularly when dealing with distance and time.

Function domain

The domain of a function is the complete set of possible input values that the function can accept and still provide a real and valid output. For functions like \(g(y)=\frac{y+2}{\sqrt{y-10}} \), determining the domain involves analyzing where the function could potentially break down.

• First, focus on any restrictions imposed by the denominator. As in the exercise, ensure the expression under the square root and in the denominator never equals zero or becomes negative.
• Solving \( y - 10 > 0 \) shows \( y > 10 \). This means only when \( y \) is greater than 10 does the function remain defined and gives real outputs.

Each function has its unique restrictions, but tackling them the right way assures you define a clear domain. A well-defined domain allows you to predict exactly where a function operates correctly, crucial for problem-solving across math and applied sciences.