Answer: B and C.
Explanation:
A function will have an inverse, only and only if every value of y can be linked to only one value x.
A) Here we have an absolute value:
We know that for example:
I5I = 5 and I-5I = 5.
Then an absolute value function can not have an inverse function.
B) one way of having a function that is invertible or not, is see if the function is increasing or decreasing.
If a function is, for example, increasing, then as the value of x increases, also does the value of y.
Then we will never see that a single value of y is related to two or more different values of x.
Then an exponential function, which is an increasing function, will have an inverse that is also a function.
C) A cubic function that has a root in (0,0)
The cubic function that has only one root at x = 0 is the function:
f(x) = a*x^3 + b*x
where a and b are real numbers of the same sign.
Now, we can see at the first derivate of f(x) to see if it is strictly increasing or strictly decreasing.
f'(x) = 3*a*x^2 + b
Ok, as x^2 is always positive, we can know that the sign of f'(x) will always be the same, so this function is strictly increasing or strictly decreasing (it depends on the signs of a and b).
Then this function has an inverse that is also a function.