f(x) = 5x+3 show that
show that it be is bijective
Answers
Answer:
f(x)=f(y)⇒x=yf(x)=f(y)⇒x=y
Or in English: if f(x)=f(y)f(x)=f(y), then x=yx=y
Suppose f(x)=f(y)f(x)=f(y) which means
3x−5=3y−53x−5=3y−5 so 3x=3y3x=3ywhich means x=yx=y.
Therefore, f(x)f(x) is one-to-one.
Now we need to show f(x)f(x) is onto RR
This means that R=Rng(f)R=Rng(f). Now the
range of f is the set of values that f maps to from the
domain. Now, we need to show that
R⊆Rng(f)R⊆Rng(f)(we know that
Rng(f)⊆RRng(f)⊆R since f is a function)
So, suppose y∈Ry∈R and let
x=y+53x=y+53. Since y∈Ry∈R
that means x∈Rx∈R. Now consider
f(x)f(x).
f(x)=3(y+53)−5f(x)=3(y+53)−5
f(x)=y+5−5=yf(x)=y+5−5=y. This means y∈Rng(f)y∈Rng(f);
thus, R⊆Rng(f)R⊆Rng(f) and
Rng(f)⊆RRng(f)⊆R which together
mean R=Rng(f)R=Rng(f); thus, f is onto.
Since f is one-to-one and onto, it is by definition a
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