Sets, Relations & Functions
Nature of Derived Function
nta_pyq_2024_apr
Grade 11
Question:
Let $f(x)=\begin{cases}-a & \text{if } -a\leq x\leq0\\ x+a & \text{if } 0<x\leq a\end{cases}$ where $a>0$ and $g(x)=\dfrac{f(|x|)-|f(x)|}{2}$. Then the function $g:[-a,a]\to[-a,a]$ is:
neither one-one nor onto.
onto.
both one-one and onto.
one-one.
Step-by-Step Solution
Key Concept: Compute $g(x)$ piecewise using $f(|x|)$ and $|f(x)|$. From the graph: $g(x)$ takes value $-a/2$ for $x\in[-a,0]$ and $0$ for $x\in(0,a]$... not covering all of $[-a,a]$.
$g$ is neither one-one nor onto.
Correct Answer: 1