R D Sharma Solutions for Chapter: Functions, Exercise 1: EXERCISE 3.1

Let $f:R^{+}\to R$, where $R^{+}$ is the set of all positive real numbers, be such that $f\left(x\right)={\log }_{e}x$. Determine the image set of the domain of given $f$.

Let $f:R^{+}\to R$, where $R^{+}$ is the set of all positive real numbers, be such that $f\left(x\right)={\log }_{e}x$. Determine $\{x:f(x)=-2\}$.
 

If $h$ is a function defined from $R$ to $R$ as follows: $h\left(x\right)=x^2+1$

Find the range of the function.

Let $X=\{\mathrm{1,2},\mathrm{3,4}\}$ and $Y=\{\mathrm{1,5},\mathrm{9,11,15,16}\}$

Determine whether the following set is a function from $X$ to $Y$:

$f_2=\left\{\left(\mathrm{1,1}\right),\left(\mathrm{2,7}\right),\left(\mathrm{3,5}\right)\right\}$

Let $X=\{\mathrm{1,2},\mathrm{3,4}\}$ and $Y=\{\mathrm{1,5},\mathrm{9,11,15,16}\}$

Determine whether the following set is a function from $X$ to $Y$:

$f_3=\left\{\left(\mathrm{1,5}\right),\left(2,9\right)\left(\mathrm{3,1}\right),\left(\mathrm{4,5}\right),\left(\mathrm{2,11}\right)\right\}$. 

Let $A=\left\{p,q,r,s\right\}$ and $B=\left\{\mathrm{1,2},3\right\}$. Which of the following relations from $A$ to $B$ is not a function?

If $f\left(x\right)=x^2$, find $\frac{f\left(1.1\right)-f\left(1\right)}{\left(1.1\right)-1}$.

If $f\left(x\right)={\left(x-a\right)}^2{\left(x-b\right)}^2$, $f\left(a+b\right)$$={\left(ab\right)}^k$. Find $k$.