i know what they mean but i what to know how do you prove and disprove if possible a function is:
injective
subjective
bijective
please please gave every possible way you know for e.g the line test and the f(x)=f(x) thingy
any help is appreciated :)
injective
subjective
bijective
please please gave every possible way you know for e.g the line test and the f(x)=f(x) thingy
any help is appreciated :)
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To prove that a function is injective, as you have said, we will need to show using arbitrary values a, b which are in the given domain of f(x) that if f(a) = f(b), then a = b.
Alternatively, we can show that if a =/= b, then f(a) =/= f(b).
The general method here is just to substitute a and b into f(x) and slowly work out both sides to show that both sides are equal or not equal.
To disprove that a function is injective, we just need to show that there exists two values in the domain a and b, where a =/= b, and f(a) = f(b). Note that these two values are real numbers in the domain, not arbitrary values.
The general method here is to substitute in the two values you have picked into f(x) and show that f(x) will have the same value for the two different values you have chosen.
To prove that a function is surjective, you must show that for each b in the codomain, there exists a in the domain of f(x) such that f(a) = b. Both a and b are arbitrary values in this case.
The general method here is to let y = f(x), rearrange the entire function into x = f^-1 (y), and argue that x is in the domain of f(x).
This can be a bit more complicated, so let me show you an example:
Let f: R->R; f(x) = x + 2
Let y = f(x). We have:
y = x + 2
x = y - 2
Since y and 2 are both real numbers, x must be a real number as well.
This means that x is in the domain of f(x).
Thus, for any y in the codomain, there must be an x in the domain such that f(x) = y.
Thus, f(x) is surjective.
Now, to prove that f(x) is not surjective, you just need to find a value in the codomain and claim that there is no a in the domain such that f(a) will result in that particular value. Note that you will need to use trial and error, or observation, to find such a value.
To prove that a function is bijective, you must show that it is both injective and surjective; no shortcuts here.
To prove that a function is not bijective, you just need to show that it it either not injective, or not surjective.
Alternatively, we can show that if a =/= b, then f(a) =/= f(b).
The general method here is just to substitute a and b into f(x) and slowly work out both sides to show that both sides are equal or not equal.
To disprove that a function is injective, we just need to show that there exists two values in the domain a and b, where a =/= b, and f(a) = f(b). Note that these two values are real numbers in the domain, not arbitrary values.
The general method here is to substitute in the two values you have picked into f(x) and show that f(x) will have the same value for the two different values you have chosen.
To prove that a function is surjective, you must show that for each b in the codomain, there exists a in the domain of f(x) such that f(a) = b. Both a and b are arbitrary values in this case.
The general method here is to let y = f(x), rearrange the entire function into x = f^-1 (y), and argue that x is in the domain of f(x).
This can be a bit more complicated, so let me show you an example:
Let f: R->R; f(x) = x + 2
Let y = f(x). We have:
y = x + 2
x = y - 2
Since y and 2 are both real numbers, x must be a real number as well.
This means that x is in the domain of f(x).
Thus, for any y in the codomain, there must be an x in the domain such that f(x) = y.
Thus, f(x) is surjective.
Now, to prove that f(x) is not surjective, you just need to find a value in the codomain and claim that there is no a in the domain such that f(a) will result in that particular value. Note that you will need to use trial and error, or observation, to find such a value.
To prove that a function is bijective, you must show that it is both injective and surjective; no shortcuts here.
To prove that a function is not bijective, you just need to show that it it either not injective, or not surjective.