Functions
Properties of Functions
MMTS_Full_Test_15
Grade 12
Question:
Number of correct statements: I) $f,g:\mathbb{R}\to\mathbb{R}$ with $f(x)=x$ if rational, $0$ if irrational and $g(x)=0$ if rational, $x$ if irrational. Then $f-g$ is one-one and onto. II) $f(x)=\sin x+\cos ax$ is periodic only if $a$ is rational. III) If $f:A\to B$ and $g:B\to C$, $g\circ f:A\to C$ is one-one, then $f$ must be one-one.
Step-by-Step Solution
Key Concept: Analyze each statement
I) True ($f-g=x$ for all $x$: bijection). II) True ($\sin x+\cos ax$ periodic iff $2\pi/a/2\pi=1/a$ rational iff $a$ rational). III) True (standard). 3 correct.
Correct Answer: 4