Q 47. Determine the domains of the fun... [FREE SOLUTION]

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Chapter 12: Q 47. (page 932)

Determine the domains of the functions in Exercises 47鈥�56, and find where the functions are continuous.
$f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$

Short Answer

Expert verified

The function $f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$ is continuous on the set $\{(x, y) 鈭� f^{2} : x^{2} 鈮� y^{2}\}$ .

Step by step solution

Step 1. Given information.

We have given expression: $f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$

Determine the domains of the functions.

Consider the function: $g : f^{2} 鈫� f$ . Then the domain of the function is

$D o m a i n (g) = \{(x, y) 鈭� f^{2} : g (x, y) i s d e f i n e d\}$

Since the rational function $f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$ is defined for all $(x, y) 鈭� f^{2}$ such that

$x^{2} - y^{2} 鈮� 0 x^{2} 鈮� y^{2}$

The domain of the function isrole="math" localid="1653241952198" $D o m a i n (f) = \{(x, y) 鈭� f^{2} : x^{2} 鈮� y^{2}\}$
.

Step 3. To find continuous of the function.

Since $x^{2}$ and $x^{2} - y^{2}$ being a polynomial function of two variable is continuous for every point on $f^{2}$ .

The rational function is continuous where all those points where $\frac{x^{2}}{x^{2} - y^{2}}$ is defined.

The rational function is discontinuous only at the points where $x^{2} - y^{2} = 0$ that is $x = y$ .

Hence the given function $f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$ is continuous on the set $\{(x, y) 鈭� f^{2} : x^{2} 鈮� y^{2}\}$ .

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91影视

Determine the domains of the functions in Exercises 47鈥�56, and find where the functions are continuous.
$f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$

Short Answer

Step by step solution

Step 1. Given information.

Determine the domains of the functions.

Step 3. To find continuous of the function.

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91影视

Determine the domains of the functions in Exercises 47鈥�56, and find where the functions are continuous.f(x,y)=x2x2-y2

Short Answer

Step by step solution

Step 1. Given information.

Determine the domains of the functions.

Step 3. To find continuous of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Applied Mathematics

Discrete Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Determine the domains of the functions in Exercises 47鈥�56, and find where the functions are continuous.
$f (x, y) = \frac{x^{2}}{x^{2} - y^{2}}$