Let f: R - R be a function given by f(x) = ax + b for all x element R - Maths - Relations and Functions

Question

Let f: R - R be a function given by f(x) = ax + b for all x
element R Find the constants a and b such that fof =
IR
Let f : R - R and g : R - R be two functions such that fog(x) =
sin x2 and gof (x) = sin2x Then find f(x) and
g(x)

Shridhar Kaushik · Accepted Answer

(1) We have â€“ $f o f = I_{R} \Rightarrow f o f (x) = I_{R} (x) \forall x \in R \Rightarrow f (f (x)) = x \forall x \in R [∵ IR (x)= x \forall x \in R] \Rightarrow f (a x + b) = x \forall x \in R \Rightarrow a (a x + b) + b = x \forall x \in R \Rightarrow (a^{2} - 1) x + a b + b = 0 \forall x \in R Now (a^{2} - 1) x + a b + b = 0 is an identity in x ∴ a^{2} - 1 = 0 and a b + b = 0 \Rightarrow a^{2} = 1 and a b + b = 0 \Rightarrow a = \pm 1 and b (a + 1) = 0 when a = 1, b (1 + 1) = 0 \Rightarrow b = 0 when a = - 1, b (a + 1) = 0 for all b \in R ∴ a = - 1 and b can take any real value$ Hence, either a = 1 and b = 0 or a = â€“ 1 and b can take any real value (2) We have â€“ $f o g (x) = \sin x^{2} and g o f (x) = \sin^{2} x \Rightarrow f (g (x)) = \sin (x)^{2} and g (f (x)) = (\sin x)^{2} \Rightarrow f (x) = \sin x and g (x) = x^{2} [\begin{matrix} ∵ when we take f (x) = \sin x and g (x) = x^{2}, f o g (x) = f (g (x)) = f (x^{2}) = \sin x^{2}] \\ and g o f (x) = g (f (x)) = g (\sin x) = (\sin x)^{2} \end{matrix}]$

Let f: R - R be a function given by f(x) = ax + b for all x element R. Find the constants a and b such that fof = IR Let f : R - R and g : R - R be two functions such that fog(x) = sin x2 and gof (x) = sin2x. Then find f(x) and g(x).

Let f: R - R be a function given by f(x) = ax + b for all x element R. Find the constants a and b such that fof = I_R

Let f : R - R and g : R - R be two functions such that fog(x) = sin x² and gof (x) = sin²x. Then find f(x) and g(x).