Question

Assertion :Let $f:[8\pi ,9\pi ]\to [-1,1],f\left(x\right)=cosx,$ then
Statement-1 : $f^{-1}(-1)=9\pi$ because Reason: Statement-2 : $f^{-1}\left(x\right)=10\pi -co{s}^{-1}x\forall x\infty [-1,1]$

• A. Statement-1 is true, Statement-2 is true and Statement-2 is correct explanation for Statement-1
• B. Statement-1 is true, Statement-2 is true and Statement-2 is NOT correct explanation for Statement-1
• C. Statement-1 is true, Statement-2 is false
• D. Statement-1 is false, Statement-2 is true

Answer 1

The correct option is C Statement-1 is true, Statement-2 is false

f(x)=cos(x).
$f^{-1}(-1)=9\pi$
$cos\left(\pi \right)=cos\left(9\pi \right)$
domain of ${\cos }^{-1}x$ is [-1,1]
range of ${\cos }^{-1}xis[0,\pi ]$
in the question given the domain specified is $[9\pi ,10\pi ]$
always the inverse function value should be in the given range.
statement 1 is true.
statement 2 is false.
$f^{-1}\left(x\right)=8\pi +{\cos }^{-1}x$