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Byju's Answer

Standard XII

Mathematics

Properties of Composite Function

Let f , g , h...

Question

Let $f, g, h$ be three bijections. If $f (x) = x^{3} + 3 x + 1,$ $g (f (x)) = x$ and $h (g (g (x))) = x,$ then the value of $h (0)$ is

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Solution

$f (x) = x^{3} + 3 x + 1, f (0) = 1, f (1) = 5$
$g (f (x)) = x \Rightarrow g (f (0)) = 0 \Rightarrow g (1) = 0 . . . (1)$
and $g (f (1)) = 1 \Rightarrow g (5) = 1 . . . (2)$

We have
$h (g (g (x))) = x$
Put $x = 5$
$\Rightarrow h (g (g (5))) = 5$
$\Rightarrow h (g (1)) = 5$
$\Rightarrow h (0) = 5$

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Let f,g,h be three bijections. If f(x)=x3+3x+1, g(f(x))=x and h(g(g(x)))=x, then the value of h(0) is

f(x)=x3+3x+1, f(0)=1, f(1)=5 g(f(x))=x⇒g(f(0))=0⇒g(1)=0 ...(1) and g(f(1))=1⇒g(5)=1 ...(2) We have h(g(g(x)))=x Put x=5 ⇒h(g(g(5)))=5 ⇒h(g(1))=5 ⇒h(0)=5

Let $f, g, h$ be three bijections. If $f (x) = x^{3} + 3 x + 1,$ $g (f (x)) = x$ and $h (g (g (x))) = x,$ then the value of $h (0)$ is

$f (x) = x^{3} + 3 x + 1, f (0) = 1, f (1) = 5$
$g (f (x)) = x \Rightarrow g (f (0)) = 0 \Rightarrow g (1) = 0 . . . (1)$
and $g (f (1)) = 1 \Rightarrow g (5) = 1 . . . (2)$

We have
$h (g (g (x))) = x$
Put $x = 5$
$\Rightarrow h (g (g (5))) = 5$
$\Rightarrow h (g (1)) = 5$
$\Rightarrow h (0) = 5$