Functions · Mathematics · AP EAPCET

Which of the following function are odd?

I. $f(x)=x\left(\frac{e^x-1}{e^x+1}\right)$

II. $f(x)=k^x+k^{-x}+\cos x$

III. $f(x)=\log \left(x+\sqrt{x^2+1}\right)$

Define the function, $f, g$ and $h$ from $R$ to $R$ such that $f(x)=x^2-1, g(x)=\sqrt{x^2+1}$ and $h(x)= \begin{cases}0, \text { if } & x \leq 0 \\ x, \text { if } & x \geq 0\end{cases}$ consider the following statements

(i) fog is invertible

(ii) $h$ is an identify function

(iii) $f \circ g$ is not invertible

(iv) $(h \circ f \circ g) x=x^2$

Then, which one of the following is true ?

Let $a > 1$ and $0 < \mathrm{b} < 1$. If $f: R \rightarrow[0,1]$ is defined by $f(x)=\left\{\begin{array}{ll}a^x, & -\infty < x < 0 \\ b^x, & 0 \leq x < \infty\end{array}\right.$, then $f(x)$ is

$$f(x)=\log \left(\left(\frac{2 x^2-3}{x}\right)+\sqrt{\frac{4 x^4-11 x^2+9}{|x|}}\right) \text { is }$$

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 $$4\left(\alpha^2+\beta^2\right)=$$

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

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

$$f(x)=\sin x+\cos x \cdot g(x)=x^2-1$$, 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

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

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

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

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

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)

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

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

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 $$R$$