Mathematics

Problem

Let

f:\mathbb{R} \to (-\infty,4]

be defined by

f(x)=4-\mathrm{e}^{2x}

. a. (i) Express

x

in terms of a general element

y

of the codomain by solving

y=f(x)

. b. (ii) Using the result of part (i), prove that

f

is surjective onto

(-\infty,4]

. c. (iii) Would the same rule

f(x)=4-\mathrm{e}^{2x}

be surjective if the codomain is declared as

\mathbb{R}

? Justify your answer.