Function question. Is this correct?

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Homework Statement


h:x → 4-x2, x E ℝ

show that it is not surjective(not onto ℝ)

The Attempt at a Solution



Since the line tests fail.

y= 4-x^2

x= √(4-y) = 2√-y

A root of a negative number is not possible so f(x) is not surjective onto R
 

Answers and Replies

  • #2
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Homework Statement


h:x → 4-x2, x E ℝ

show that it is not surjective(not onto ℝ)

The Attempt at a Solution



Since the line tests fail.
If it fails the horizontal line test (meaning that a horizontal line intersects two or more points on the graph), that means that the function is not one-to-one. What did you mean?

Do you understand the definition of "onto" (or surjective)?
y= 4-x^2

x= √(4-y) = 2√-y
What are you doing here? When you solve for x, you should get two values; namely, x = ±√(4 - y). Also, it is NOT true that √(4-y) = 2√-y.
A root of a negative number is not possible so f(x) is not surjective onto R
 
  • #3
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I meant when I did the line tests it looked like the function was injective and surjective

and hm... I'm not too sure about the 2nd part now umm
x=±√(4-y) if y is like 3 x is a real number..
 
  • #4
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For the function y = x2 - 4 to be onto the real numbers, it must be true that any choice of y is paired with some value of x.

Have you graphed this equation? That would probably give you a good idea about whether it is onto the reals. That wouldn't be proof, but it would get you thinking the right way.
 
  • #5
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I only sketched a graph,
 
  • #6
Dick
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I only sketched a graph,

Ok, then what's a value in R that x^2-4 can never equal?
 
  • #7
haruspex
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x=±√(4-y) if y is like 3 x is a real number..
Sure, but to be surjective on ℝ the inverse must have a solution for all real y. Does it?
 
  • #8
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No y is not surjective for the positive Real numbers though..

"Ok, then what's a value in R that x^2-4 can never equal? "

Umm I'm not sure..
 
  • #9
HallsofIvy
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Homework Statement


h:x → 4-x2, x E ℝ

show that it is not surjective(not onto ℝ)

The Attempt at a Solution



Since the line tests fail.

y= 4-x^2

x= √(4-y) = 2√-y

A root of a negative number is not possible so f(x) is not surjective onto R
Almost correct. First, as Mark44 said, it should be "[itex]\pm[/itex]" and surely you know that "4- y" is NOT "4 times -y"! [itex]x= \pm\sqrt{4- y}[/tex]. Now, can you find a value of y so that 4- y< 0?
 
  • #10
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Can't 5 make it <0?
 
  • #11
Dick
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Can't 5 make it <0?

If you mean there is no real value of x such that 4-x^2=5, that would be correct.
 
  • #12
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But aren't negative numbers be.. real?
 
  • #13
haruspex
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Can't 5 make it <0?
Of course, but HofI wasn't suggesting you could not. Halls just said, find such a value of y. Now, having found it, what value of x will be mapped to this y?
 
  • #14
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If x=-4 that could make f(x) < 0
 
  • #15
haruspex
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If x=-4 that could make f(x) < 0
You are not trying to make f(x) < 0. Go and look at Halls' post again. You were trying to make 4-y < 0. You correctly found that y=5 would do that. Now the question is whether you can find an x that makes f(x) = 5. If no such x exists then f is not surjective.
 
  • #16
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Oh well it's not surjective then because I can't think of any value... If this is the answer I'm sorry for being so difficult, I need MUCH more practice in functions..
 
  • #17
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Your notation for the function definition isn't correct, I think.

I believe you meant [itex]f : \mathbb{R} \rightarrow \mathbb{R}, x \in \mathbb{R} \mapsto 4 - x^2[/itex].

Here, [itex]\mathbb{R}[/itex] is both the domain and codomain of [itex]f[/itex].

Surjectivity is the property that the image of the domain of [itex]f[/itex], which is defined and denoted to be [itex]f[\mathbb{R}]=\{f(x) : x \in \mathbb{R}\}[/itex], equals the codomain of [itex]f[/itex].

Thus, we want to see if we can generate all the real numbers with [itex]f[/itex].

Analytically, this function is a parabola starting at [itex](0,4)[/itex] and opening down. What does this imply, then?

Also, a algebraic argument can provide a solution. Suppose [itex]y \in \mathbb{R}[/itex] is some value in the codomain of [itex]f[/itex]. Furthermore, suppose that there exists some value [itex]x \in \mathbb{R}[/itex] in the domain of [itex]f[/itex] such that [itex]f(x)=4-x^2=y[/itex]. If you solve for [itex]y[/itex], what do you discover?
 
  • #18
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Umm that Y is > or equal to 4 no matter number you use?
 
  • #19
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Umm that Y is > or equal to 4 no matter number you use?

What is your reasoning? Substitute some values for [itex]x[/itex] into [itex]f[/itex] and see what you get, or what you can't get. Does your answer change? Even better, use graphing software to visualize [itex]f[/itex], and then everything should be pretty clear. (Example, search "wolframalpha" into a search engine and type "f(x) = 4 - x^2" into the search bar on the website. A graph over a restricted amount of points will be generated. Personally, I would recommend solving this algebraically, but analytically is totally fine, too.)
 
  • #20
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But solving it alebraically isn't it this??

y= 4-x^2
x^2= 4-y
x= sqrt(4-y)

putting in y=5 x would = sqrt(-1) so... it's not surjective as the codomain doesn't equal the range?
 
  • #21
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But solving it alebraically isn't it this??

y= 4-x^2
x^2= 4-y
x= sqrt(4-y)
The line above should be x = ±sqrt(4-y)

putting in y=5 x would = sqrt(-1) so... it's not surjective as the codomain doesn't equal the range?
In other words, if y = 5, there is no real value of x for which 4 - x2 = 5.

As you said earlier(or at least, alluded to), the range of the function y = 4-x2 is the set {y | y ≤ 4}. This should be enough to convince anyone that this function is not onto the reals.
 
  • #22
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But solving it alebraically isn't it this??

y= 4-x^2
x^2= 4-y
x= sqrt(4-y)

putting in y=5 x would = sqrt(-1) so... it's not surjective as the codomain doesn't equal the range?

Do you mean letting [itex]y=5[/itex]? If so, then yes, you get [itex]\pm \sqrt{-1}[/itex], which is not a real number, and therefore there does not exist any [itex]x[/itex] in the domain of [itex]f[/itex] such that [itex]f(x)=5[/itex]. Consequently, the range of [itex]f[/itex] does not include [itex]5[/itex], for example. That is enough to show that surjectivity is not held by [itex]f[/itex], as the set of real numbers (the domain) does not equal the set of real numbers missing [itex]5[/itex] (the codomain) (of course, it missed more than just [itex]5[/itex]).

Recall you said [itex]f[/itex] attains all numbers greater than or equal to [itex]5[/itex] only--do you see the error now?

Also, recall that taking the square root always requires one to put a plus or minus [itex]\pm[/itex] sign in front of the root. Why? Well, for any number [itex]x[/itex] such that the square root of it is defined, we have both [itex](\sqrt{x})^2 = x[/itex] and [itex](-\sqrt{x})^2 = x[/itex].
 

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