Question

The function f(x) defined on a real number has the property that $f\left( {f\left( x \right)} \right) \cdot \left( {1 + f\left( x \right)} \right) = - f\left( x \right)$ for all x in the domain of f. If number 3 is in the domain and the range of f, compute the value of f(3).
A.$\dfrac{{ - 3}}{4}$
B.$\dfrac{{ - 3}}{2}$
C.$\dfrac{2}{3}$
D.$\dfrac{{ - 2}}{3}$

Answer 1

In this question, we need to determine the value of the function at the value of x as 3 such that $f\left( {f\left( x \right)} \right) \cdot \left( {1 + f\left( x \right)} \right) = - f\left( x \right)$ should be satisfied. For this, we will use the simple arithmetic operations and the rearrangements of the given expression to get the result.

Complete step-by-step answer:
The given expression is $f\left( {f\left( x \right)} \right) \cdot \left( {1 + f\left( x \right)} \right) = - f\left( x \right)$
Taking the term $\left( {1 + f\left( x \right)} \right)$ to the right-hand side, we get

$$\Rightarrow f\left( {f\left( x \right)} \right) \cdot \left( {1 + f\left( x \right)} \right) = - f\left( x \right) \\\ \Rightarrow f\left( {f\left( x \right)} \right) = \dfrac{{ - f\left( x \right)}}{{1 + f\left( x \right)}} - - - - (i) \\\$$

According to the question, the function f(x) has been defined on a real number, so let the function f(x) be ‘x’ such that x belongs to the set of the real numbers.
Substituting the value of the function f(x) as x in the equation (i), we get

$$\Rightarrow f\left( {f\left( x \right)} \right) = \dfrac{{ - f\left( x \right)}}{{1 + f\left( x \right)}} \\\ \Rightarrow f(x) = \dfrac{{ - x}}{{1 + x}} - - - - (ii) \\\$$

Now, we have been asked to determine the value of the function at x=3. So, substitute x=3 in the equation (ii) to evaluate the value of the function.

$$\Rightarrow f(x) = \dfrac{{ - x}}{{1 + x}} \\\ \Rightarrow f(3) = \dfrac{{ - 3}}{{1 + 3}} \\\$$

Simplifying the above equation, we get

$$\Rightarrow f(3) = \dfrac{{ - 3}}{{1 + 3}} \\\ = \dfrac{{ - 3}}{4} \\\$$

Hence, the value of the function f(x) at x=3 is given as $\dfrac{{ - 3}}{4}$ such that the function f(x) has been defined on real numbers and $f\left( {f\left( x \right)} \right) \cdot \left( {1 + f\left( x \right)} \right) = - f\left( x \right)$.
Option A is correct.

Note: It is interesting to note here that the range and the domain of the function include 3 and so as our results should come within the given range only. Real numbers consist of zero (0), the positive and negative integers (-3, -1, 2, 4), and all the fractional and decimal values in between (0.4, 3.1415927, 1/2). Real numbers are divided into rational and irrational numbers.