 # 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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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