# Determining the domain and range of multi-variable function

• cathal84
In summary, the function f(x,y) = 1/(y^2-x) has a domain of all real numbers except when y^2-x = 0, and the range is all real numbers except 0. The range can be defined as the set of real numbers z such that z = 1/(y^2-x).
cathal84

## Homework Statement

f(x,y) = 1/y^2-x

find the domain of f.

Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f.

## Homework Equations

I know that the domain of the function is anywhere that the function is defined.

## The Attempt at a Solution

in the case of this question i can see that the function is going to be undefined where y^2-x =0
since anything over 0 is undefined.

I believe that for the range part of the question it is essentially saying, define the range such that x and y are real numbers but please correct me if I'm wrong.

When i used to be dealing with functions such as f(x) = x+1, in that case i knew that the domain was where the function was defined over the x-axis and i knew that the range was where the function was defined over the y-axis.

But since the domain now is including X and Y i have no idea by what they mean by finding the range.

So if someone could help me define the range of this function and show me the steps to defining the range of any other multi-variable like this one i would greatly appreciate it!
This question is purely for study purposes for exam in start of January so feel free to go into as much detail as you want if you feel it would aid understanding. Thanks again!

Last edited:
cathal84 said:

## Homework Statement

f(x,y) = 1/y^2-x
You need to write the right side as 1/(y2 - x).
What you wrote means ##\frac 1 {y^2} - x##, which isn't what you meant, based on your work below.
cathal84 said:
find the domain of f.

Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f.

## Homework Equations

I know that the domain of the function is anywhere that the function is defined.

## The Attempt at a Solution

in the case of this question i can see that the function is going to be undefined where y^2-x =0
since anything over 0 is undefined.

I believe that for the range part of the question it is essentially saying, define the range such that x and y are real numbers but please correct me if I'm wrong.
The range is the set of real numbers z such that ##z = \frac 1 {y^2 - x}##. Obviously 0 is not in the range of this function. Are there any other values that aren't in the range?
cathal84 said:
When i used to be dealing with functions such as f(x) = x+1, in that case i knew that the domain was where the function was defined over the x-axis and i knew that the range was where the function was defined over the y-axis.

But since the domain now is including X and Y i have no idea by what they mean by finding the range.

So if someone could help me define the range of this function and show me the steps to defining the range of any other multi-variable like this one i would greatly appreciate it!
This question is purely for study purposes for exam in start of January so feel free to go into as much detail as you want if you feel it would aid understanding. Thanks again!

cathal84 said:

## Homework Statement

f(x,y) = 1/y^2-x

find the domain of f.

Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f.

You wrote
$$f(x,y) = \frac{1}{y^2} - x,$$
at least when we read your expression using standard parsing rules. If you mean something else you need to use parentheses.
If you mean
$$f(x,y) = \frac{1}{y^2-x},$$
then write it in plain text as f(x,y) = 1/(y^2-x).

Mark44 said:
You need to write the right side as 1/(y2 - x).
What you wrote means ##\frac 1 {y^2} - x##, which isn't what you meant, based on your work below.
The range is the set of real numbers z such that ##z = \frac 1 {y^2 - x}##. Obviously 0 is not in the range of this function. Are there any other values that aren't in the range?
Ray Vickson said:
You wrote
$$f(x,y) = \frac{1}{y^2} - x,$$
at least when we read your expression using standard parsing rules. If you mean something else you need to use parentheses.
If you mean
$$f(x,y) = \frac{1}{y^2-x},$$
then write it in plain text as f(x,y) = 1/(y^2-x).
Mark44 said:
You need to write the right side as 1/(y2 - x).
What you wrote means ##\frac 1 {y^2} - x##, which isn't what you meant, based on your work below.
The range is the set of real numbers z such that ##z = \frac 1 {y^2 - x}##. Obviously 0 is not in the range of this function. Are there any other values that aren't in the range?
ah yes sorry ill know for next time to put the brackets in thanks for the advice.
I don't see why any other numbers would not be in the range besides 0 but then again I'm just looking at where the function is undefined but that's how you define the domain, so i am not sure, think i am just confusing myself. what exactly am i looking to do differently when defining the range?

## What is the domain of a multi-variable function?

The domain of a multi-variable function is the set of all possible input values for the function. In other words, it is the independent variable or variables that can be plugged into the function to produce an output.

## What is the range of a multi-variable function?

The range of a multi-variable function is the set of all possible output values for the function. In other words, it is the dependent variable or variables that are produced by plugging in the input values.

## How do you determine the domain of a multi-variable function?

To determine the domain of a multi-variable function, you must first identify all the independent variables in the function. Then, you must consider any restrictions or limitations on those variables. Finally, you can determine the range by finding the set of all possible values that satisfy those restrictions.

## How do you determine the range of a multi-variable function?

To determine the range of a multi-variable function, you must first identify all the dependent variables in the function. Then, you must consider any restrictions or limitations on those variables. Finally, you can determine the range by finding the set of all possible values that satisfy those restrictions.

## Why is it important to determine the domain and range of a multi-variable function?

Determining the domain and range of a multi-variable function is important because it helps us understand the behavior of the function and identify any limitations or restrictions on its inputs and outputs. This information is crucial in many areas of mathematics, science, and engineering.

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