Functions
Odd-even decomposition of a function
Grade Class 12
Question:
Let $f(x)=x^2+x$ be written as $f(x)=g(x)+h(x)$ where $g(x)$ is an odd function and $h(x)$ is an even function. Then $g(xy)+h\!\left(\dfrac{x}{y}\right)$ equals
$xy+\dfrac{x^2}{y^2}$
$xy-\dfrac{x^2}{y^2}$
$\dfrac{x^2}{y^2}-xy$
none of these
Step-by-Step Solution
Key Concept: $g(x)=(f(x)-f(-x))/2=x$ (odd part); $h(x)=(f(x)+f(-x))/2=x^2$ (even part). Then $g(xy)+h(x/y)=xy+(x/y)^2$.
$g(xy)+h(x/y)=xy+x^2/y^2$.
Correct Answer: 1