Determine whether each of these functions is a bijection from R to R. a) f (x)

Joni Kenny 2021-09-07 Answered

Determine whether each of these functions is a bijection from R to R.

a) \(f (x) = −3x + 4\)

b) \(\displaystyle{f{{\left({x}\right)}}}=−{3}{x}^{{{2}}}+{7}\)

c) \(f (x) = \frac{x + 1}{x + 2}\)
\(\displaystyle{d}{)}{f{{\left({x}\right)}}}={x}^{{{5}}}+{1}\)

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Expert Answer

Ayesha Gomez
Answered 2021-09-08 Author has 10704 answers

Part a-
\(f(x)=-3x+4\) is a bijection.
It is one-to-one as \(\displaystyle{f{{\left({x}\right)}}}={f{{\left({y}\right)}}}\Rightarrow-{3}{x}+{4}=-{3}{y}+{4}\Rightarrow{x}={y}\)
It is onto as \(\displaystyle{f{{\left({\frac{{{4}-{x}}}{{{3}}}}\right)}}}={x}\)
Part b-
\(f(-x)=f(x)\).

Hence the function is not a bijection.
Part c-
There is no real number x such that \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{x}+{1}}}{{{x}+{2}}}}={1}\). Hence the function is not a bijection.
Part d-
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{{5}}}+{1}\) is a bijection.
It is a strictly increasing function.
Result:
Part a - Yes
Part b - NO
Part c - No
Part d - Yes

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