Let f - N - Y be a function defined as f x - 4x- 3- where Y - y - N-y - 4x- 3 for some x- N - Show that f is invertible- Find the inverse-

Calculate: g : Y → N Given: f (x) = 4x+ 3 Let f (x) = y y = 4x+ 3 4x= y 3 x = y 34 Let g (y) = y 34 where g : Y → N Solve to prove gof = I_N. gof = g (f(x)) ⇒ ...

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Standard XII

Mathematics

Derivative

Let f : N → Y...

Question

Let $f : N \to Y$ be a function defined as $f (x) = 4 x + 3$ , where $Y = {y \in N : y = 4 x + 3 for some x \in N}$ . Show that $f$ is invertible. Find the inverse.

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Solution

Calculate: $g : Y \to N$
Given: $f (x) = 4 x + 3$
Let $f (x) = y$
$y = 4 x + 3$
$4 x = y - 3$
$x = \frac{y - 3}{4}$
Let $g (y) = \frac{y - 3}{4}$ where $g : Y \to N$

Solve to prove $g o f = I_{N}$ .
$g o f = g (f (x))$
$\Rightarrow g o f = g (4 x + 3)$
$\Rightarrow g o f = \frac{(4 x + 3) - 3}{4}$
$\Rightarrow g o f = \frac{4 x}{4} = x = I_{N} . . . (1)$

Solve to prove $f o g = I_{y}$ .
$f o g = f (g (x))$
$\Rightarrow f o g = f (\frac{y - 3}{4})$
$\Rightarrow f o g = 4 (\frac{y - 3}{4}) + 3$
$\Rightarrow f o g = y - 3 + 3$
$\Rightarrow f o g = y = I_{y} . . . (2)$
From (1) and (2)
$g o f = I_{N}$ and $f o g = I_{y}$
$\Rightarrow f$ is invertible, so
Inverse of $f = g (y) = \frac{y - 3}{4}$

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Derivative

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Let f:N→Y be a function defined as f(x)=4x+3, where Y={y ∈ N:y=4x+3 for some x∈ N}. Show that f is invertible. Find the inverse.

Let $f : N \to Y$ be a function defined as $f (x) = 4 x + 3$ , where $Y = {y \in N : y = 4 x + 3 for some x \in N}$ . Show that $f$ is invertible. Find the inverse.