Problem 11
Question
Determine the domain of each function of two variables. $$ f(x, y)=\sqrt{y-3 x} $$
Step-by-Step Solution
Verified Answer
The domain is all \((x, y)\) such that \(y \geq 3x\).
1Step 1: Identify the Key Component
The function given is a square root function: \( f(x, y) = \sqrt{y - 3x} \). The expression inside the square root must be non-negative for the function to be defined in the set of real numbers.
2Step 2: Set the Condition for the Domain
Since the expression inside the square root, \( y - 3x \), must be non-negative, we set the inequality: \( y - 3x \geq 0 \). This ensures that the square root is defined.
3Step 3: Solve the Inequality
To find the domain, solve the inequality \( y - 3x \geq 0 \). This can be rewritten as \( y \geq 3x \). The domain consists of all \((x, y)\) pairs where this condition holds.
4Step 4: Express the Domain
The solution to the inequality \( y \geq 3x \) describes the domain of the function. In interval notation, the domain can be expressed as the set \((x, y) \in \mathbb{R}^2\) such that \(y \geq 3x\).
Key Concepts
Domain of a FunctionInequalitiesReal Numbers
Domain of a Function
The domain of a function is a crucial concept in mathematics. It determines where the function is defined and can take values from. For multivariable functions like ours, which involve two variables, the domain consists of all possible pairs of values the function can accept without causing any mathematical errors.
In the function given as an example, \( f(x, y) = \sqrt{y - 3x} \), we need the expression inside the square root, \( y - 3x \), to be non-negative. This is important because square roots of negative numbers aren't defined in the real number system, leading to complex (or imaginary) numbers instead.
To find out where the function is defined, we set up the condition \( y - 3x \geq 0 \). This inequality shows the valid input combinations of \(x\) and \(y\) that ensure \(f(x, y)\) remains real. Therefore, the domain is all pairs \((x, y)\) where \( y \geq 3x \). This set describes a half-plane on the Cartesian plane.
In the function given as an example, \( f(x, y) = \sqrt{y - 3x} \), we need the expression inside the square root, \( y - 3x \), to be non-negative. This is important because square roots of negative numbers aren't defined in the real number system, leading to complex (or imaginary) numbers instead.
To find out where the function is defined, we set up the condition \( y - 3x \geq 0 \). This inequality shows the valid input combinations of \(x\) and \(y\) that ensure \(f(x, y)\) remains real. Therefore, the domain is all pairs \((x, y)\) where \( y \geq 3x \). This set describes a half-plane on the Cartesian plane.
Inequalities
Inequalities are expressions that indicate the relative sizes or order of two values. In the problem involving our function, the inequality \( y - 3x \geq 0 \) plays a central role in defining the domain. Solving this inequality helps us understand the region where the function is valid.
The inequality can be interpreted as a statement that \( y \) is greater than or equal to \( 3x \). This forms a boundary line on a graph, which in this case is the line \( y = 3x \).
To solve the inequality:
The inequality can be interpreted as a statement that \( y \) is greater than or equal to \( 3x \). This forms a boundary line on a graph, which in this case is the line \( y = 3x \).
To solve the inequality:
- Recognize that \( y = 3x \) is the line of equality, where the function changes from being undefined (below the line) to defined (on and above the line).
- All points above this line, including the line itself, satisfy \( y \geq 3x \), making the square root non-negative and the function defined.
Real Numbers
The realm of real numbers includes all the numbers that can exist on the number line. Real numbers are essential in determining if a mathematical expression, especially in functions involving square roots, is valid.
Square root functions are only defined for non-negative inputs in the real number system. If the input to the square root is negative, we obtain complex numbers, which fall outside the scope of real numbers.
By focusing on real numbers and their properties:
Square root functions are only defined for non-negative inputs in the real number system. If the input to the square root is negative, we obtain complex numbers, which fall outside the scope of real numbers.
By focusing on real numbers and their properties:
- We ensure the outputs of our square root functions remain within the real number system.
- This keeps the function \( f(x, y) = \sqrt{y - 3x} \) applicable and usable in real-world contexts where imaginary numbers are not practical.
Other exercises in this chapter
Problem 11
Find \(f_{x}\) and \(f_{y}\). $$f(x, y)=e^{x y}$$
View solution Problem 11
Find the relative maximum and minimum values. $$ f(x, y)=4 x^{2}-y^{2} $$
View solution Problem 12
Evaluate. $$ \int_{0}^{1} \int_{-1}^{x}\left(x^{2}+y^{2}\right) d y d x $$
View solution Problem 12
Find \(f_{x}\) and \(f_{y}\). $$f(x, y)=e^{2 x y}$$
View solution