Functions - Calculus PYQ | Mathematics Exams

The real valued function \(f(x)=\frac{x}{e^x-1}+\frac{x}{2}+1\) defined on \(R /\{0\}\) is

The domain of the function \(f(x)=\frac{1}{[x]-1}\), where \([x]\) is greatest integer function of \(x\) is

Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\frac{4^x}{4^x+2}\), what is the value of \(f\left(\frac{1}{4}\right)+2 f\left(\frac{1}{2}\right)+f\left(\frac{3}{4}\right)\) is equal to

Given, the function \(f(x)=\frac{a^x+a^{-x}}{2},(a>2)\), then \(f(x+y)+f(x-y)\) is equal to

Let \(f: R \rightarrow R\) and \(g: R \rightarrow R\) be defined by \(f(x)=2 x+1\) and \(g(x)=x^2-2\) determine \((g \circ f)(x)\) is equal to

If \({({x^2} + 5x + 5)^{x + 5}} = 1\), then the number of integers satisfying this equation is

If \(f\) is a function defined on \((0,1)\) by \(f(x)=\min \{x-[x],-x-[x]\}\), then \((f \circ f o f o f)(x)\) is equal to \(\rightarrow([\cdot]\) greatest integer function)

Which statement among the following is true?

(i) the function \(f(x)=x|x|\) is strictly increasing on \(R-\{0\}\).

(ii) the function \(f(x)=\log _{(1 / 4)} x\) is strictly increasing on \((0, \infty)\).

(iii) a one-one function is always an increasing function.

(iv) \(f(x)=x^{1 / 3}\) is strictly decreasing on \(R\)

If \(\frac{x^4}{(x-1)(x-2)}=f(x)+\frac{A}{x-1}+\frac{B}{x-2}\), then

If \(f(10-x)=3 x^2+4 x-5\) and \(f(x)=p x^2+q x+r\), then \(p+q+r\) is equal to

If \(f: R \rightarrow R\) and \(g: R \rightarrow R\) are two functions defined by \(f(x)=a x+b(a \neq 0), \forall x \in R\) and \(g(x)=c x^3+d(c \neq 0), \forall x \in R\), then \((f \circ g)^{-1}(x)\) is equal to

If \(f\) is the greatest integers function defined on \(R\) as \(f(x)=[x]\) and \(g\) is the modulus function defined on $R$ as \(g(x)=|x|\), then the value of \((g \circ f)\left(\frac{-5}{3}\right)\) is

Let \(f(x)=(x+2)^2-2, x \geq-2\). Then, \(f^{-1}(x)\) is equal to

Let \(f(x)=x^3\) and \(g(x)=3^x\), then the quadratic equation whose roots are solutions of the equation \((f \circ g)(x)=(g \circ f)(x)\) (for \(x \neq 0\)) is

\(f(x)=\sin x+\cos x \cdot g(x)=x^2-1\), then \(g(f(x))\) is invertible if

If \(f: z \rightarrow z\) is defined by \(f(x)=x^9-11 x^8-2 x^7+22 x^6+x^4 -12 x^3+11 x^2+x-3, \forall x \in z\), then \(f(11)\) is equal to

The domain of the real valued function \(f(x)=\sin \left(\log \left(\frac{\sqrt{4-x^2}}{1-x}\right)\right.\) is

If \(f(x)=\sqrt{2-x^2}\) and \(g(x)=\log (1-x)\) are two real valued functions, then the domain of the function \((f+g)(x)\) is

The range of the real valued function \(f(x)=\sqrt{\frac{x^2+2 x+8}{x^2+2 x+4}}\) is

Let \(f: R-\left\{\frac{-1}{2}\right\} \rightarrow R\) be defined by \(f(x)=\frac{x-2}{2 x+1}\). If \(\alpha\) and \(\beta\) satisfy the equation \(f(f(x))=-x\), then \(4\left(\alpha^2+\beta^2\right)=\)