Question

Which one of the following pairs are equal functions?
A) $\ln {x^2},2\ln x;x > 0$
B) $\dfrac{{{x^2}}}{x},x$
C) $\left| {{x^2}} \right|,{x^2}$
D) $\left| {{x^3}} \right|,{x^3}$

Answer 1

All the four pairs are of well-defined functions on their respective domains so we have to verify each and every function for equality. We say that two functions are the same if their domains are equal and for each and every point it takes the same values.

Complete step-by-step answer:
There is no direct connection between any of the pairs so we have no choice than to check for each and every pair of the function so that we can find the functions with equal values.
So, we will start from the first pair.
A) $\ln {x^2},2\ln x;x > 0$
It is already given that the function is well defined in the domain.
We know that for the logarithms $a\ln x = \ln {x^a}$ .
Therefore, $\ln {x^2} = 2\ln x$
Thus, this pair has equal functions.
B) $\dfrac{{{x^2}}}{x},x$
The first term here is $\dfrac{{{x^2}}}{x}$, now this term is well-defined only when $x \ne 0$.
Whereas the other function is always well-defined.
Thus, domains of both the functions are different.
Thus, this is not the pair of equal functions.
C) $\left| {{x^2}} \right|,{x^2}$
This one is easy. We know that mod function is well defined and it is always equal to the respective square function.
Therefore, the third pair is of equal functions.
D) $\left| {{x^3}} \right|,{x^3}$
We will illustrate this with an example.
First of all, both the functions are well-defined over entire real numbers.
Let us take $x = - 1$ .
Then $\left| {{x^3}} \right| = 1$ but ${x^3} = - 1$ .
Therefore, we found the point where the functional values are different.
Thus, this is not the pair of the equal functions.

Therefore, A) and C) are equal function pairs.

So, the correct answer is “Option A and C”.

Note: Here we used different tactics to show the equality and inequality of the function. Showing that two functions are unequal is easier than showing equality. Each example uses a different technique. Recall the definitions of the domains and range of the function to get a good grasp.