Consider f: R\(\to\)R given by f(x) = 4x+3. Show that f is invertible. Find the inverse of f.
f: R \(\to\) R is given by,
f(x) = 4x + 3
One-one:
Let f(x) = f(y).
\(\implies\)4x+3 = 4y+3
\(\implies\) 4x = 4y
\(\implies\)x = y.
∴ f is a one-one function.
Onto:
For y ∈ R, let y = 4x + 3.
\(\implies\)x = \(\frac {y-3}{4}\) ∈R
Therefore, for any y ∈ R, there exists x = \(\frac {y-3}{4}\) ∈R such that
f(x) = f\((\frac {y-3}{4})\) = 4\((\frac {y-3}{4})\)+3 = y.
∴ f is onto.
Thus, f is one-one and onto and therefore, f−1 exists.
Let us define g: R\(\to\) R by g(y) = \((\frac {y-3}{4})\).
Now (g0f)(x) = g(f(x)) = g(4x+3) = \(\frac {(4x+3)-3}{4}\)=x.
(fog)(y) = f(g(y)) = f\((\frac {y-3}{4})\) = 4\((\frac {y-3}{4})\)+3 = y-3+3 = y.
therefore gof = fog = IR
Hence, f is invertible and the inverse of f is given by
f-1(y) = g(y) = \(\frac {y-3}{4}\).
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is: