Mock Test 1 — Functions

The domain of definition of the real function $f(x) = \frac{1}{\sqrt{x - |x|}}$ is:

If the mapping $f: \mathbb{R} \to \mathbb{R}$ is defined by the rule $f(x) = 2x + 3$, then its inverse function $f^{-1}(x)$ is:

Determine if the function $f(x) = x \cdot \left(\frac{a^x + 1}{a^x - 1}\right)$ is even, odd, or neither:

The fundamental period of the standard periodic function $f(x) = \cos(4x)$ is equal to:

The total number of unique bijections that can be defined from a finite set of $3$ elements to itself is:

The range of the fraction expression $f(x) = \frac{1}{x^2 + 2}$ is bounded by which interval?

If $f(x) = x^2$ and $g(x) = \sqrt{x}$, then the composite expression $(f \circ g)(x)$ is identical to $x$ only over which domain?

Replacing $x$ with $x - 3$ inside a function's argument shifts its graph geometrically:

The output value of the greatest integer function expression $\lfloor -2.1 \rfloor$ is equal to:

For a binary operation defined by $a * b = a + b - ab$, the identity element $e$ is:

Find the number of integers in the range of the function $f(x) = 3\sin x + 4$.

If the fundamental period of the function $f(x) = \sin(kx)$ is $\frac{\pi}{3}$, find the value of the positive integer $k$.

Evaluate the value of the composite function iteration $f(f(2))$ if $f(x) = \frac{x+2}{x-1}$.

Assertion (A): The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^3$ is a bijective function.
Reason (R): The derivative of the function is non-negative ($f'(x) = 3x^2 \ge 0$), making it strictly increasing (one-one), and its range spans from $-\infty$ to $\infty$ (onto).

Assertion (A): The graph of $y = f(-x)$ is obtained by reflecting the curve $y = f(x)$ across the horizontal x-axis.
Reason (R): Reflecting a function horizontally across the vertical y-axis replaces every input coordinate $x$ with its negative counterpart $-x$.