Functions practice · Medium

Option A (4) is correct because solving the inequality derived from the area condition yields four possible integer values for alpha.

Option B (3) is correct because the number of real values satisfying the equation is found by determining the number of intersection points between the graphs of f(x) and g(x), which is 3.

Option C ([1/7, 7]) is correct because the range is found by determining the minimum and maximum values of the function using calculus.

Based on the provided information, the simplified form of $f(x)=|x-2|+|2-x|,-3 \leq x \leq 3$ is $\left\{\begin{array}{cc}2 x, & 2 \leq x \leq 3 \\ 4, & -2 \leq x<2 \\ -2 x, & -3 \leq x<-2\end{array}\right.$, which is option A.

Option C (A bijection) is correct because the function is both one-to-one and onto.

Option D (\mathbb{R}-\{-1,1\}) is correct because the function is not differentiable at x = -1 and x = 1.

Option C ($2^{8}$) is correct because the total number of reflexive relations on a set with $n$ elements is $2^{n(n-1)}$, and for a set with 4 elements, this is $2^{4(4-1)} = 2^{12}$. However, the explanation provided in the original answer is incorrect. The correct number of reflexive relations is $2^{n^2 - n} = 2^{4^2 - 4} = 2^{12}$. The provided answer $2^4$ is incorrect.

Option B (-1) is correct because it is the value of f(0) obtained by substituting x=0 and x=1 in the given functional equation and solving the resulting system of equations.

Option B ($\{1\}$) is correct because the expression $x^2 - 2x + 3$ is always positive, so its sign is always 1.

Option B ($f(x)=x(\frac{{a^x-1}}{{a^x+1}})$) is correct because $f(-x) = f(x)$, which is the definition of an even function.

Option B is correct because it represents the inverse function $f^{-1}(x)$ obtained by solving for x in terms of y and swapping x and y.

Option D (None of the above) is correct because the relation R, defined by $x^2 + y^2 = 9$, includes the pairs (0, 3), (0, -3), (3, 0), and (-3, 0). The domain and range are both {-3, 0, 3}, so all the statements are true, making "None of the above" the correct answer.

Option B ($(\operatorname{{sgn}} x) \sqrt{{\frac{{|x|}}{{1-|x|}}}}$) is correct because it represents the inverse function of the given function, considering the sign of x.

Option C (5/3) is correct because by substituting x = 1 and x = -1/2 into the given equation, we obtain a system of two equations that can be solved for f(2).

Option A ([{1,\sqrt 2 }]) is correct because it represents the range of the secant function given the restricted domain of its argument.

Option A is correct because the range of the function ${x^3} - 12x$ for $-3 \le x \le 3$ is $[-16, 16]$.

Option C (One-one and into) is correct because the exponential function $f(x) = e^x$ is injective (one-to-one) but not surjective (onto) when mapping from R to R.

Option A (4) is correct because there are four ordered pairs (x, y) that satisfy the given equations involving f(x, y) and g(x, y).

Option B is correct because f{g(x)} = e^(log_e(x)) = x and g{f(x)} = log_e(e^x) = x, so f{g(x)} = g{f(x)}.

Option C is correct because the function h(n) is not defined when n = 0, as it would involve division by zero.

Option A ($\frac{{2a+b}}{{2(a^2-b^2)}}$) is correct because it is the value of f(2) obtained by solving the system of equations formed by substituting x=1 and x=-1/2 into the given functional equation.

Option D is correct because the range of sin(x) - sqrt(3)cos(x) - 1 is [-1, 3] when the function is onto.

Option B (Assertion is correct, reason is correct; reason is not a correct explanation for assertion) is correct because both the assertion and reason are true statements, but the reason does not explain the assertion.

Option D (Statement- 1 is true, Statement- 2 is true. Statement- 2 is a correct explanation for Statement-1) is correct because the function is a bijection and the set of x values where f(x) = f^-1(x) is {0, -1}.

The correct answer is D ((-4, 0) ∪ (0, ∞)) because it represents the values of x for which the square root is defined (x + 4 > 0) and the denominator is non-zero (x ≠ 0).

Option B ([2, ∞)) is correct because the condition |f(x) + φ(x)| = |f(x)| + |φ(x)| is satisfied when both f(x) and φ(x) are non-negative.

Option B ($g\left( x \right) = x + \frac{1}{x},x \in \left( {0,\infty } \right)$) is correct because it is not injective, as different values of x can produce the same output.

Option D ($[1, \infty]$) is correct because the base of a logarithm must be positive and not equal to 1, and the argument must be positive.

Option B ($f(8)=f(9)$) is correct because it is derived from the given functional equation by substituting appropriate values for x and y.

Option D ($\frac{1-2^{-10}}{3^{10}-2^{-10}}$) is correct because it is the solution to the equation $f(g(x)) = x$ after substituting the given functions.