Question

f(x) = 5x+3 show that
show that it be is bijective​

Answer 1

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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