(a) Statement- 1 is True, Statement- 2 is True; Statement- 2...

(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 If $f(x)=\sin ^{-1}(\log x)$, then $f(1)=0$ and Statement 2 graph of $f(x)=\log _{t} x, 0<a<1$ is.

(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
If $f(x)=\sin ^{-1}(\log x)$, then $f(1)=0$
and
Statement 2
graph of $f(x)=\log _{t} x, 0<a<1$ is.

Key Concepts

Inverse Trigonometric Functions

These functions, such as the arcsine (sin?¹), serve as the inverse of the standard trigonometric functions. They are defined with particular ranges—in the case of sin?¹, the output typically lies between -?/2 and ?/2—which ensure that each input within the domain produces a unique output. This is essential when evaluating expressions like sin?¹(0).

Logarithm Functions

Logarithm functions are the inverses of exponential functions. One key property is that the logarithm of 1 is always 0, regardless of the base (provided the base is positive and not equal to 1). They obey specific rules regarding their domains and behaviors, making them fundamental in analyzing functions that involve logarithmic components.

Composition of Functions

Function composition involves applying one function to the results of another. In expressions where one function is nested inside another, such as sin?¹(log x), it is crucial to consider the domain restrictions of both functions to ensure that the overall function is well-defined. The output of the inner function must lie within the domain of the outer function.

Graphical Properties of Logarithmic Functions with Bases Between 0 and 1

When the base of a logarithmic function is between 0 and 1, the graph of the function is decreasing, as opposed to the increasing behavior seen with bases greater than 1. This characteristic impacts the shape of the graph and is important when interpreting or comparing functions based on their graphical behavior.

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