Select problems from miscellaneous exercise

In this article we will look at solutions of a few selected problems from miscellaneous exercise on chapter 2 of NCERT.

Let $f=\{(x,{\displaystyle \frac{x^2}{1+x^2}}):x\in \mathbb{R}\}$‍ be a function from $\mathbb{R}$‍ into $\mathbb{R}$‍. Determine the range of $f$‍.

Solution:

Remember that range is the set of all outputs of a function. Here,

The input $x$‍ can take any real value. The question is, as $x$‍ varies across $\mathbb{R}$‍, what values does the function take?

Look at the expression ${\displaystyle \frac{x^2}{1+x^2}}$‍. Remember $x^2\ge 0$‍. So, this expression

See that the numerator and denominator are always positive, so the expression is always positive. Also the denominator is always greater than the numerator, so the expression is always less than $1$‍.

Let's verify what we found above by checking the output of the function at various values of $x$‍.

The lowest value $f(x)$‍ takes is $0$‍ at $x^2=0$‍. It increases towards $1$‍ as $x^2$‍ increases. But however hard we try, we never reach exactly $1$‍ because the denominator always stays greater than the numerator.

The function takes all values between $0$‍ and $1$‍, including $0$‍ but excluding $1$‍.

The range of $f$‍ is $[0,1)$‍.

Let $f$‍ be the subset of $\mathbb{Z}\times \mathbb{Z}$‍ defined by $f=\{(ab,a+b):a,b\in \mathbb{Z}\}$‍. Is $f$‍ a function from $\mathbb{Z}$‍ to $\mathbb{Z}$‍? Justify your answer.

Proving something is not a function is generally much more easier than proving something is a function.

So first, let's try to prove that $f$‍ is not a function. To do so, we need to match at least one input to two or more outputs.

Consider the pair $(ab,a+b)$‍. Here $ab$‍ is the input and $a+b$‍ is the output.

Pick a input. Let's say $12$‍. Now let's try to match this input to two (or more) outputs.

See that we got multiple outputs for the same input $12$‍.

Therefore, $f$‍ is not a function.