Finding the Domain of Function f

Oct 13, 2025 by windailys.com 127 views

**Finding the Domain and Range of Functions f and g**

Understanding the Basics

When dealing with functions, it's essential to understand the concepts of domain and range. The domain of a function is the set of all possible input values (x-values) that the function can accept, while the range is the set of all possible output values (y-values) that the function can produce.

In this article, we will explore the domain and range of two given functions, f and g, and provide a step-by-step guide on how to find them.

Function f:

The function f is defined as:

f(x) = (x+2)/(4x-9)

Function g:

The function g is defined as:

g(x) = √(x-15)

Finding the Domain of Function f

To find the domain of function f, we need to determine the values of x that make the denominator (4x-9) non-zero.

Step 1: Find the values of x that make the denominator zero

Set the denominator equal to zero and solve for x:

4x - 9 = 0

x = 9/4

Step 2: Determine the intervals where the function is defined

Since the denominator cannot be zero, we need to find the intervals where the function is defined. The function is defined for all real numbers except x = 9/4.

Step 3: Write the domain in interval notation

The domain of function f is:

(-∞, 9/4) ∪ (9/4, ∞)

Finding the Range of Function f

To find the range of function f, we need to determine the possible output values (y-values) of the function.

Step 1: Analyze the behavior of the function

The function f is a rational function, and its behavior can be analyzed by looking at the numerator and denominator.

Step 2: Determine the vertical asymptote

The vertical asymptote of the function is x = 9/4, which is the value that makes the denominator zero.

Step 3: Determine the horizontal asymptote

The horizontal asymptote of the function is y = 1/4, which is the ratio of the leading coefficients of the numerator and denominator.

Step 4: Determine the range

The range of function f is all real numbers except y = 1/4.

Finding the Domain of Function g

To find the domain of function g, we need to determine the values of x that make the expression inside the square root non-negative.

Step 1: Find the values of x that make the expression inside the square root non-negative

x - 15 ≥ 0

x ≥ 15

Step 2: Determine the intervals where the function is defined

The function is defined for all real numbers x ≥ 15.

Step 3: Write the domain in interval notation

The domain of function g is:

[15, ∞)

Finding the Range of Function g

To find the range of function g, we need to determine the possible output values (y-values) of the function.

Step 1: Analyze the behavior of the function

The function g is a square root function, and its behavior can be analyzed by looking at the expression inside the square root.

Step 2: Determine the minimum value of the function

The minimum value of the function is g(15) = √(15-15) = 0.

Step 3: Determine the range

The range of function g is all real numbers greater than or equal to 0.

Conclusion

In this article, we have explored the domain and range of two given functions, f and g. We have provided a step-by-step guide on how to find the domain and range of each function, and have discussed the key concepts and techniques used in the process.

Q&A

Q: What is the domain of function f?

A: The domain of function f is (-∞, 9/4) ∪ (9/4, ∞).

Q: What is the range of function f?

A: The range of function f is all real numbers except y = 1/4.

Q: What is the domain of function g?

A: The domain of function g is [15, ∞).

Q: What is the range of function g?

A: The range of function g is all real numbers greater than or equal to 0.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to analyze the function and determine the values of x that make the function defined and the possible output values (y-values) of the function.

Q: What are the key concepts and techniques used in finding the domain and range of a function?

A: The key concepts and techniques used in finding the domain and range of a function include analyzing the behavior of the function, determining the vertical and horizontal asymptotes, and determining the intervals where the function is defined.

Q: Can you provide more examples of finding the domain and range of functions?

A: Yes, we can provide more examples of finding the domain and range of functions. Please let us know what type of functions you would like to see examples of, and we will do our best to provide them.