It Use the function given in Exercise 31 . (a) Find the...

It Use the function given in Exercise $31 .$ (a) Find the domain and range of the function. (b) Identify the points in the $x y$ -plane at which the function value is 0 (c) Does the surface pass through all the octants of the rectangular coordinate system? Give reasons for your answer.

It Use the function given in Exercise $31 .$
(a) Find the domain and range of the function.
(b) Identify the points in the $x y$ -plane at which the function value is 0
(c) Does the surface pass through all the octants of the rectangular coordinate system? Give reasons for your answer.

Key Concepts

Zeros of the Function

Zeros or roots of a function are the input values where the output equals zero. Finding these points involves setting the function equal to zero and solving the resulting equation, which provides the intersection points of the function with the horizontal axis in the coordinate plane.

Rectangular Coordinate System and Octants

The rectangular coordinate system, also known as the Cartesian coordinate system, divides three-dimensional space into eight regions called octants. Analyzing whether a surface passes through all the octants involves examining the sign variations of the coordinates (x, y, and z) on the surface and confirming that there are points on the surface in each of these regions.

Domain

The domain of a function refers to the complete set of possible input values (usually x-values) for which the function is defined. It involves determining any restrictions such as division by zero or taking even roots of negative numbers that could lead to undefined expressions, ensuring that only valid inputs are considered.

Range

The range of a function is the set of all possible outputs (usually y-values or function values) that can result from using the numbers in the domain. Identifying the range typically involves analyzing the behavior of the function, including its limits, extrema, or the effect of transformations applied to the function.

Transcript

00:01 So in this question, we have a function f of xy is equal to negative 4x over x squared plus y squared plus 1.

00:15 And we want to know first about the domain and range of this function.

00:22 So thinking about my domain to start, when would i have restrictions on my domain? well, i would have restrictions on my domain whenever.

00:34 My denominator was equal to zero, right? whenever x squared plus y squared plus one were to equal zero.

00:44 Equivalently, i'm saying i have restrictions on my domain whenever x squared plus y squared is equal to negative one.

00:54 But x squared, that's always non -negative and y squared.

01:02 That's always non -negative as well.

01:05 And so the sum of two non -negatives cannot equal negative 1.

01:10 And so x squared plus y squared equaling negative 1, that's never happening.

01:18 My domain is in fact going to be all of two -dimensional space.

01:27 How about the range of this function? well, my range, i believe, is going to be the set of all real numbers from negative infinity to positive.

01:37 Positive infinity.

01:39 Looking back at the original function, f of x, y, notice that my denominator, x squared, plus y squared, plus one, this guy is always going to be positive, right? that's always greater than zero.

01:57 But this means that if my numerator is negative, my f of x, y is negative...

Please give Ace some feedback