Intersecting Bivariate Functions: Is x=a the Only Point of Intersection?

  • Context: High School 
  • Thread starter Thread starter Saracen Rue
  • Start date Start date
  • Tags Tags
    Intersection
Click For Summary
SUMMARY

The discussion centers on the intersection of bivariate functions, specifically whether the functions ##f(x,a)## and ##f(a,x)## always intersect at the point ##x=a##. It is established that for an intersection to occur, the two functions must be equal, leading to the conclusion that while other points of intersection may exist, the point ##x=a## is a trivial solution. The participants clarify that the equality condition simplifies to ##f(x,x) = f(x,x)##, indicating that the question may have already been addressed in previous discussions.

PREREQUISITES
  • Understanding of bivariate functions
  • Knowledge of function equality and intersection points
  • Familiarity with mathematical notation and terminology
  • Basic comprehension of real-valued functions
NEXT STEPS
  • Explore the properties of bivariate functions in greater depth
  • Research the concept of function intersections and their implications
  • Study examples of non-trivial intersections in bivariate functions
  • Investigate the role of symmetry in function intersections
USEFUL FOR

Mathematicians, students studying advanced calculus, and anyone interested in the properties of bivariate functions and their intersections.

Saracen Rue
Messages
150
Reaction score
10
Will the bivariate function ##f(x,a)## always intersect ##f(a,x)## at the point ##x=a## given that ##f## is a real, defined function? (other points of intersection can exist but are not relevant for this question)
 
Physics news on Phys.org
Hey Saracen Rue.

If you have an intersection then it means that the two things are equal.

So if you had a bivariate function with points (a,b) and (c,d) then an intersection happens when f(a,b) = f(c,d)

You have specified [if I read correctly] that a=x, b=a', c=a', d=x and a'=x meaning that you have f(x,x) = f(x,x) which is trivial.

Are we missing something here?
 

Similar threads

Undergrad Why Isn't the Intersection of Two Lines a 1D Manifold?
  • · Replies 21 ·
Replies
21
Views
4K
MHB Proving Topology in X: A Look at Union & Intersection
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad A non-empty intersection of closures of level sets implies discontinuity
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Exercise 2.23 in Folland's real analysis text
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Uniform closure of algebra of bounded functions is uniformly closed
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Intersection of a closed convex set
  • · Replies 1 ·
Replies
1
Views
2K
High School Topological neigborhoods that are not intervals in the real line?
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Homotopy Class vs Fundamental Group.
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Question about lifting of a function
  • · Replies 2 ·
Replies
2
Views
3K
High School What kind of definition is this?
  • · Replies 23 ·
Replies
23
Views
2K