Inverse Functions

Let $a>1$ be a real number and $f\left(x\right)={\log }_a{x}^2$ for $x>0 .$ If $f^{-1}$ is the inverse function of $f$, and $b$ and $c$ are real numbers, then $f^{-1}\left(b+c\right)$ is equal to

If $X$ and $Y$ are two non-empty sets where $f:X\to Y$, is function is defined such that

$f\left(c\right)=f\left\{f\left(x\right):x \epsilon C\right\}$ for $D\subseteq X$

and $f^{-1}\left(D\right)=\left\{x:f\left(x\right) \epsilon D\right\}$ for $D\subseteq Y$,

for any $A\subseteq Y$ and $B\subseteq Y$, then

Find the formula for the given function. Further, explain why it has an inverse function and hence find the formula of its inverse function.

The volume of a certain type of wood in terms of the weight: 

Find the formula for the given function. Further, explain why it has an inverse function and hence find the formula of its inverse function.

The price $P$ you pay in Happy Shop in terms of the regular price $R$ if there is a sign that says "Today you pay $15\%$ less for every single item".

Find the formula for the given function. Further, explain why it has an inverse function and hence find the formula of its inverse function.

The number of pounds in weight, $p$, relating to a weight in kilograms, $k$. Note that $1$kg is equivalent to $2.2$ pounds.

If $\text{g}$ is the inverse of a function $f$ and $f^{\prime }\left(\text{x}\right)=\frac{1}{1+{\text{x}}^5}\text{, }\text{then }{\text{g}}^{\prime }\left(\text{x}\right)$ is equal to 

The diagram shows the graph of a function $f$ from $A$ to $B$.

Find $f^{-1}\left(-1\right)$.

The diagram shows the graph of a function $f$ from $A$ to $B$.

Find $f^{-1}\left(1\right)$.

Describe the difference between a linear function that has an inverse and one that does not.

Describe the difference between a linear function that has an inverse and one that does not.

Describe the difference between a linear function that has an inverse and one that does not.

Describe the difference between a linear function that has an inverse and one that does not.

 A function $f\left(x\right)$ is defined by $f\left(x\right)=3x-10$, the range of function $f\left(x\right)$ is $5<f\left(x\right)<50$.

State the range of the inverse function.

 A function $f\left(x\right)$ is defined by $f\left(x\right)=3x-10$.

Find the inverse function, $f^{-1}\left(x\right)$ of given function.

Determine whether the following function have an inverse function. Give a reason.

$f\left(x\right)=\sqrt{x}$

Determine whether the following function have an inverse function. Give a reason.

$f\left(x\right)=3$

Determine whether the following function have an inverse function. Give a reason.

$f\left(x\right)=5x+1$

Determine whether the following function have an inverse function. Give a reason.

$f\left(x\right)=x^2$

The one-to-one function $f$ maps $A=\{x:x\ge 0\}$ onto $B=\{y:-1\le y\le 2\};f\left(1\right)=0.5$ and $f\left(0\right)=2$. Solve the equation $f^{-1}\left(x\right)=0$.