What is the range of the quadratic function f(x,y) = (xy-x^2, xy-y^2)?

In summary, the range of a quadratic function is the set of all possible output values that the function can produce. It can be determined by analyzing the shape and position of the parabola and can be negative, infinite, or finite. In real-life applications, the range can be used to analyze and model situations and determine maximum or minimum values.
  • #1
JohnSimpson
92
0
Consider the map [tex]f : \mathbb{R}^2 \rightarrow \mathbb{R}^2[/tex]
defined by
[tex](x,y) \mapsto (xy-x^2, xy-y^2)[/tex]

I'm interested in figuring out the range of this function, but I keep thinking myself in circles. What would be a systematic method for approaching something like this?
 
Mathematics news on Phys.org
  • #2
Writing (u, v) for the new co-ordinates, you could look at a linear combination of these, u+m.v say, and find the extremal points as a function of m. This will give you a parametric equation describing the boundary.
In the present case, u+v is interesting.
 

Related to What is the range of the quadratic function f(x,y) = (xy-x^2, xy-y^2)?

1. What is the range of a quadratic function?

The range of a quadratic function is the set of all possible output values, or y-values, that the function can produce. It can be written as a set of numbers or a mathematical inequality.

2. How is the range of a quadratic function determined?

The range of a quadratic function can be determined by analyzing the shape and position of the parabola. If the parabola opens upwards, the range will be all real numbers greater than or equal to the y-coordinate of the vertex. If the parabola opens downwards, the range will be all real numbers less than or equal to the y-coordinate of the vertex.

3. Can the range of a quadratic function be negative?

Yes, the range of a quadratic function can be negative. This occurs when the parabola opens downwards and the y-coordinate of the vertex is a negative number. The range will then be all real numbers less than or equal to this negative y-coordinate.

4. Is the range of a quadratic function always finite?

No, the range of a quadratic function can also be infinite. This occurs when the parabola opens upwards and the y-coordinate of the vertex is positive infinity, or when the parabola opens downwards and the y-coordinate of the vertex is negative infinity.

5. How can I use the range of a quadratic function in real-life applications?

The range of a quadratic function can be used to analyze and model real-life situations, such as projectile motion or the profit of a business. It can also help determine the maximum or minimum value of a variable in a given scenario.

Similar threads

B Cosecant function: notation for the domain and range -- Am I right?
    • General Math
Replies
6
Views
877
MHB Is $f(x) = xf(1)$ the only solution to the given functional equation?
    • General Math
Replies
3
Views
972
I What are the applications of inverses of vector functions?
    • General Math
Replies
17
Views
2K
MHB Quadratic equations intersaction point is minimum instead of roots
    • General Math
Replies
2
Views
764
MHB Given f(xy) = f(x)/y and f(500) = 3, find f(600)
    • General Math
Replies
2
Views
2K
MHB Quadratic functions: Which of these quadratic functions has exactly one x -intercept
    • General Math
Replies
3
Views
2K
MHB Determine all real valued differentiable functions f(x)+f(y)=f(xy)
    • General Math
Replies
4
Views
1K
MHB What Is the Domain and Range of y=cos(3(x - 45°)) +2?
    • General Math
Replies
3
Views
725
MHB Solving Function Expressions h(x): (x+a)2 +b & h-1 Range
    • General Math
Replies
7
Views
1K
B Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
    • General Math
Replies
3
Views
809

Hot Threads

Recent Insights

Back
Top