Can You Solve One-to-One Functions Algebraically?

  • Context: High School 
  • Thread starter Thread starter AznBoi
  • Start date Start date
  • Tags Tags
    Functions
Click For Summary

Discussion Overview

The discussion revolves around the algebraic determination of whether a function is one-to-one, specifically focusing on the function f(x) = (3x + 4)/5. Participants explore methods to prove one-to-one characteristics, including algebraic manipulation and the use of calculus, while also addressing misconceptions related to other functions like f(x) = x^2.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants inquire about the process of solving for one-to-one functions algebraically, questioning what substitutions for x should be made.
  • Others suggest starting with the equality f(a) = f(b) and manipulating it to show a = b as a proof of one-to-one nature.
  • There is a discussion on using calculus to demonstrate that the derivative is always positive, implying the function does not double back on itself.
  • Some participants express confusion about the implications of squaring values, particularly in the context of the function f(x) = x^2, which is not one-to-one.
  • Concerns are raised about the definition of one-to-one functions, with some participants suggesting their understanding may differ from others in the thread.
  • A counter-example is presented to illustrate that a single instance can disprove a general claim about one-to-one functions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the methods for proving one-to-one functions, and there are multiple competing views regarding the definitions and implications of one-to-one characteristics, particularly in relation to specific examples like f(x) = x^2.

Contextual Notes

Some participants express uncertainty about the algebraic steps necessary to demonstrate one-to-one properties, and there are unresolved questions regarding the definitions and examples of one-to-one functions.

AznBoi
Messages
470
Reaction score
0
How do you solve one-to-one functions algebraically?

Problem: f(x) (3x+4)/5

What are you suppose to substitue for x?

I know that if f(a)=f(b), a=b...
 
Mathematics news on Phys.org
What do you mean solve?

If f(x) = (3x+4)/5, then you have your function. If you plug in an x, you can solve for f(x) by doing basic math. Given what f(x) is, you can do something like this:

5*f(x) = 3x + 4

5f(x) - 4=3x

(5f(x)-4)/3=x
 
It says: Determine algebraically whether the function is one-to-one. How do I do that?
 
Oh, I get it. You want to show that if you plug two distinct values, x1 and x2 into f(x), the values that are returned either are not equal to each other, or x1 and x2 are the same (that's the definition of one to one)
 
So do you plug 2 and -2. Their negatives? or what two numbers do you plug in. I plugged in 2 and -2 and I got 2 and -2/5. What am I doing wrong?
 
AznBoi said:
How do I do that?
Just do what it says:

I know that if f(a)=f(b), a=b...
Start by writing down f(a) = f(b).
Use algebra to derive a = b.

Proof finished.
 
Can you use calculus? If so, you can show that the slope (derivative) of the function is always positive and bounded. That means that the function cannot double-back on itself to create a second y value for any x value.

Even if you aren't supposed to use calculus, at least for this problem, the equation is the equation of a straight line, right? y = mx + b


EDIT -- Ooo. I like Hurkyl's method better!
 
what do you mean? like f(a)=3a+4/5 f(b)=3b+4/5?? a will always equal b if you do it that way wouldn't it? No I can't use calculus, I'm in pre cal xD. Can you show me how to do it?
 
AznBoi said:
what do you mean? like f(a)=3a+4/5 f(b)=3b+4/5?? a will always equal b if you do it that way wouldn't it?
If you can prove what you just said, then you've proven f is one-to-one. It's that easy.
 
  • #10
What about f(x)=x^2 It's not a one-to-one function even though f(a)=a^2 is equal to f(b)=b^2 a=b in that case but if you put -2 and 2 in f(a)=f(b) but a doesn't equal b.. So I'm confused.
 
  • #11
Can you give me some examples of functions that aren't one-by-one. I mean if you use the same function for x=a and b. aren't they alwasy equal to each other?
 
  • #12
For x^2, if a^2=b^2, then a=-b means x^2 isn't necessarily one to one
 
  • #13
What about f(x)=x^2 It's not a one-to-one function even though f(a)=a^2 is equal to f(b)=b^2 a=b in that case
Why do you think a=b in that case? You've demonstrated that's not always true... so think hard about why you would (incorrectly) believe a=b must be true here.
 
  • #14
Okay I know that -2^2 and 2^2 are both equal to 4.. So that means you can't use a and b.. cause a^2 and b^2 would be a=b if you solve it algebrically. What numbers do I need to substitue for x? How do I know that a=b or a doesn't equal b. Thats what I'm trying to figure out. =P
 
  • #15
cause a^2 and b^2 would be a=b if you solve it algebrically.
Show your work.
 
  • #16
a^2=b^2 because you square root both sides and you get a=b?
 
  • #17
a^2=b^2 because you square root both sides and you get a=b?
Nope. You get |a| = |b|.
 
  • #18
So basically anything that is to an even power is not a one-to-one function. ok this is weird. lol
 
  • #19
AznBoi said:
So basically anything that is to an even power is not a one-to-one function. ok this is weird. lol
Just for my peace of mind, could you please post the textbook definition of a one-to-one function? I think that my practical definition of a one-to-one function (any x maps to only one y) may not match what others are asking you to show.
 
  • #20
One-to-one means IF f(x) = f(y) THEN x=y.

So think about f(x) = x^2. If f(x) = f(y) is it NECESSARILY true that x=y? If not, then f is not 1-1. If so, then f is 1-1.
 
  • #21
berkeman said:
Just for my peace of mind, could you please post the textbook definition of a one-to-one function? I think that my practical definition of a one-to-one function (any x maps to only one y) may not match what others are asking you to show.

Your definition there is not of one-to-one. Merely of a function.
 
  • #22
A single "counter-example" is sufficient to show that a general statement is not true. The fact that (-1)2= 12 is sufficient to show that f(x)= x2 is not one to one.

To show that a function is one to one, you have to be more general: f(x)= x3 is one to one (as a function over the real numbers) because if a3= b3, then a= b.
 

Similar threads

Undergrad My attempt to understand horizontal transformations of functions
  • · Replies 6 ·
Replies
6
Views
3K
MHB Solve Inequality & Simul Eqns: Alg Workings Q (a,b,c)
  • · Replies 1 ·
Replies
1
Views
2K
High School What went wrong with (-x)^2=x^2?
  • · Replies 22 ·
Replies
22
Views
3K
Graduate A global version of the implicit function theorem
  • · Replies 1 ·
Replies
1
Views
1K
High School What is the notation for denoting constants in a function?
  • · Replies 2 ·
Replies
2
Views
1K
High School Question about Notation of Function
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Can the same argument be used for both radians and degrees in the sine function?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Transformation of Functions: How Do Domain and Range Change?
  • · Replies 4 ·
Replies
4
Views
2K
High School Missing out on intermediate algebraic steps
  • · Replies 23 ·
Replies
23
Views
2K
High School Inverse Functions vs Inverse Relations
  • · Replies 4 ·
Replies
4
Views
4K