Answer
(a) one to one
(b) one to one
(c) not one to one
(d) one to one
(e) not one to one
(f ) not one to one
Work Step by Step
The horizontal line process requires drawing multiple horizontal lines, $y=a,a\in R$, and observing whether any of them cross the function more than once.
A one-to-one function is one where each $y$ value is given by a single $x$ value, whereas a many-to-one function is one in which multiple $x$ values can give the same $y$ value.
If a horizontal line crosses the function more than once, it indicates that the function has more than one $x$ value that corresponds to one $y$ value.
So,
(a)$ f(x) = 3x + 2$
let's replace $x$ with $y$ and $y $ with $x $ and solve the math with respect to $x$
$x = 3y+2$
for $ x = {1,2,3,4,5}$ the value of $y $ will be $ {5,8,11,14,17}$
so, this is one to one function.
(b) $f(x) = \sqrt {x-1} $
one to one
(c) $f(x) = |x|$
not one to one. A horizontal line crosses the function more than once
(d) $f(x) = x^3$
one to one
(e) $f(x) = x^2 − 2x + 2$
not one to one. A horizontal line crosses the function more than once
(f ) $f(x) = \sin x$
not one to one. A horizontal line crosses the function more than once
