Question

$ f $ is a function such that $ f(x) < 0 $ and the graph of a new function $ g $ defined by $ g(x) = \left| {f(x)} \right| $ is a reflection of the graph of $ f $ on –
A.The y-axis
B.The x-axis
C.The line $ y = x $
D.The line $ y = - x $

Answer 1

Hint: We are given a function which is less than zero and the graph of another function is defined by absolute value of the previous function i.e. it is a reflection of the graph. Now we have to see if it is a vertical or horizontal reflection which will lead us to the answer.

Complete step-by-step answer:
Firstly we write down what is given in the question
 $ f $ is a function such that $ f(x) < 0 $
Graph of a new function $ g $ defined by $ g(x) = \left| {f(x)} \right| $ is a reflection of the graph of $ f $
From the first part we can say that the function $ f $ is a decreasing function.
From the second part we can say that $ g(x) $ equals to the absolute value of $ f(x) $ i.e.
 $ g(x) = \left| {f(x)} \right| $
Here we solve the absolute value function which is known as a piecewise function. So we have
\[
  f(x) < 0 \\
   \Rightarrow g(x) = \left| {f(x)} \right| = \left\{ {\begin{array}{*{20}{c}}
  {f(x),\;f(x) \geqslant 0} \\
  { - f\left( x \right),\;f(x) < 0}
\end{array}} \right.\; \;
 \]
We have $ f(x)\,{\text{and}}\, - f(x) $ are reflections as we can see it by the standard graph of the absolute value function

So here we have
 $ g(x) = \left| {f(x)} \right| = - f(x) $
This means by looking at the graph one can say that dashed lines show the reflection of the function which means the reflection will be on the “x-axis” in this case, and it is also called vertical reflection of a function. So our option – b is the right answer.
So, the correct answer is “Option B”.

Note: Reflection of graph works with the sign of the function i.e., when negative of the original function is taken, then the graph flips upside down. A minus sign on the whole function flips it over x-axis and a minus sign on the argument [in f(x), the part inside the bracket is argument] flips it over y-axis.