Question

If \[f(x) = {x^2} - {x^{ - 2}}\] then \[f\left( {\dfrac{1}{x}} \right)\] is equal to
A. \[f\left( x \right)\]
B. \[ - f\left( x \right)\]
C. \[\dfrac{1}{{f\left( x \right)}}\]
D. \[{\left( {f\left( x \right)} \right)^2}\]

Answer 1

Hint: Here in this question we have function f, it is a function for x. We have to substitute the \[\dfrac{1}{x}\] in the place of x and we simplify the given function. After simplification we have to compare the obtained solution with the 4 options and hence we obtain the required solution for the given question.

Complete step by step solution:
A function relates inputs to outputs. A function is a special type of relation between the variables.
Now consider the given equation or function
 \[f(x) = {x^2} - {x^{ - 2}}\]
Now we have to find the value of \[f\left( {\dfrac{1}{x}} \right)\]
Substitute the value of x as \[\dfrac{1}{x}\]
On substituting we get
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = {\left( {\dfrac{1}{x}} \right)^2} - {\left( {\dfrac{1}{x}} \right)^{ - 2}}\]
We multiply both the numerator and the denominator.
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = \dfrac{{{1^2}}}{{{x^2}}} - \dfrac{{{{(1)}^{ - 2}}}}{{{x^{ - 2}}}}\]
When 1 is multiplied twice we obtain the answer as 1 itself.
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = \dfrac{1}{{{x^2}}} - \dfrac{1}{{{x^{ - 2}}}}\]
By the law of indices we know that \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] , by considering this the above function is written as
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = {x^{ - 2}} - {x^{ - ( - 2)}}\]
On simplifying it is written as
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = {x^{ - 2}} - {x^2}\]
Take “ – “ sign as a common, the equation is written as
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = - ({x^2} - {x^{ - 2}})\]
The equation in the braces is the given function. So it can be written as
 \[ \Rightarrow f\left( {\dfrac{1}{x}} \right) = - f(x)\]
Therefore the \[f\left( {\dfrac{1}{x}} \right)\] is equal to \[ - f\left( x \right)\]
Hence the option B is the correct one.
Hence we have determined the solution for the given question.
So, the correct answer is “Option B”.

Note: The question involves the simple simplification. while simplifying we must know about the simple arithmetic operations. The law of indices \[\dfrac{1}{{{a^m}}} = {a^{ - m}}\] is used to solve the given equation. The f implies the function of some variable. Here the variable id x. The given function is the function of x.