Mock Test 2 — Functions

The domain of definition of the real-valued function $f(x) = \sqrt{\log_{0.5} (x^2 - 5x + 7)}$ is:

Let the function $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$. This function can be structurally classified as:

The fundamental period of the function $f(x) = |\sin 3x| + |\cos 3x|$ is exactly:

If $f(x) = \log_e \left(\frac{1-x}{1+x}\right)$, then the composite expression $f(x_1) + f(x_2)$ simplifies using logarithmic properties to match which option?

The total number of elements inside the range of the function $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$ is exactly:

If a function satisfies the equation $2f(x) + 3f(1/x) = x^2$ for all non-zero real numbers, then the value of $f(2)$ is equal to:

Let a graph transformation stretch a curve horizontally. Constructing the graph of $y = |x - 1| - 1$ maps its vertex to coordinates:

If the function $f(x) = \frac{4^x}{4^x + 2}$, then the value of the symmetric sum $f(x) + f(1-x)$ is:

A binary operation on positive integers is defined as $a * b = \text{HCF}(a, b)$. This operation satisfies which properties?

The range of the function $f(x) = \cos(\lfloor x \rfloor)$ over the domain $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$ contains how many distinct numerical elements?

Find the number of real roots of the transcendental equation $e^x = -x$.

Find the cardinality of the range of the fractional part expression $f(x) = \{\sin x\}$ if the domain is restricted to integers.

Let $f(x)$ be an invertible bijection such that $f(3) = 5$. Find the value of the inverse expression $f^{-1}(5)$.

Assertion (A): The product of two odd functions is always an even function.
Reason (R): Substituting $-x$ into both functions pulls out two negative signs, which multiply together to become positive ($(-1) \times (-1) = 1$).

Assertion (A): The socks-and-shoes identity statement $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$ applies to all invertible functions.
Reason (R): To undo a composite process, you must reverse the order of the individual inverse operations step-by-step.