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

Whether each of these functions is a bijection from R to R.

a) $f\left(x\right)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

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

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

The function is not a bijection as a result.
Part c-
No real number exists that is such that $f\left(x\right)=\frac{x+1}{x+2}=1$. Hence the function is not a bijection.
Part d-
$f\left(x\right)={x}^{5}+1$ 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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