Properties of Inverse Function

Q.

If $\phi \left(x\right)$ is the inverse of the function $f\left(x\right)$ and $f\left(x\right)=\frac{1}{\left(1+x^5\right)}$, then $\frac{d}{dx}\phi \left(x\right)$ is

1. $\frac{1}{\left(1+{\left\{\phi \left(x\right)\right\}}^5\right)}$

2. $1+f\left(x\right)$

3. $\left(1+{\left\{\phi \left(x\right)\right\}}^5\right)$

4. $\frac{1}{\left(1+{\left\{f\left(x\right)\right\}}^5\right)}$

Q. The inverse of a bijection is unique and also a bijection.

1. True
2. False

Q.

The inverse function of $f^{-1}\left(x\right),if{f}^{-1}\left(x\right)existsisf\left(x\right)$

1. True

2. False

Q. If $f$ is an invertible function given by $f\left(x\right)=\ln \left(x\right)+4x-8,$ then the value of $f^{-1}(-4)$ is

1. $1$
2. $0$
3. $-1$
4. $2$

Q. Consider a function $f:R-\left\{1\right\}\to R-\left\{1\right\}$ defined by $f\left(x\right)=\frac{x+a}{x-1},$ where $a$ is a real constant. If $f$ is not constant in its domain, then which of the following is/are CORRECT ?

1. domain and range of $f$ are the same
2. $\left(fof\right)\left(\frac{1}{x}\right)=\frac{1}{x}$
3. $f\left(\frac{1}{f\left(f\right(x\left)\right)}\right)-f\left(f\left(f\left(\frac{1}{x}\right)\right)\right)=0$
4. $f$ has its own inverse

Q.

The inverse function of $f^{-1}\left(x\right),if{f}^{-1}\left(x\right)existsisf\left(x\right)$

1. True

2. False

Q.

If a function f(x) is defined on [1, 4] $\to$ [1, 7] and given that f(3) = 5 and its inverse exists then find $f\left(f^{-1}\right(5\left)\right)$.

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Q. Let $f:A\to Bandg:B\to A$be two functions such that $gof=I_{A}anffog=I_B.$ Then, which of the following statement/s is/are true?

1. ${\left(f^{-1}\right)}^{-1}=f$
2. ${\left(gof\right)}^{-1}=f^{-1}o{g}^{-1}$
3. f and g are bijections
4. $g=f^{-1}$