Can You Solve One-to-One Functions Algebraically?

[AznBoi]

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...

 

[Office_Shredder]

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

 

[AznBoi]

It says: Determine algebraically whether the function is one-to-one. How do I do that?

 

[Office_Shredder]

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)

 

[AznBoi]

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?

 

[Hurkyl]

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.

 

[berkeman]

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!

 

[AznBoi]

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?

 

[Hurkyl]

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.

 

[AznBoi]

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.

 

[AznBoi]

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?

 

[Office_Shredder]

For x^2, if a^2=b^2, then a=-b means x^2 isn't necessarily one to one

 

[Hurkyl]

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.

 

[AznBoi]

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

 

[Hurkyl]

cause a^2 and b^2 would be a=b if you solve it algebrically.

Show your work.

 

[AznBoi]

a^2=b^2 because you square root both sides and you get a=b?

 

[Hurkyl]

a^2=b^2 because you square root both sides and you get a=b?

Nope. You get |a| = |b|.

 

[AznBoi]

So basically anything that is to an even power is not a one-to-one function. ok this is weird. lol

 

[berkeman]

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.

 

[Data]

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.

 

[matt grime]

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.

 

[HallsofIvy]

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.