Functions
Invertibility; one-one/many-one
MJMT_Full_Test_04
Grade 12
Question:
A function $f:\mathbb{R}\to\mathbb{R}$ is defined as $f(x)=3x^2+1$. Then $f^{-1}(x)$ is
$\dfrac{\sqrt{x-1}}{3}$
$\dfrac{1}{3}\sqrt{x-1}$
$f^{-1}$ does not exist
$\sqrt{\dfrac{x-1}{3}}$
Step-by-Step Solution
Key Concept: An inverse exists iff the function is bijective. $f(x)=3x^2+1$ is even ⟹ many-to-one.
$f(1)=f(-1)=4$, so $f$ is many-to-one. Hence $f^{-1}$ does not exist.
Correct Answer: 3