determine whether f(x)= x^4+3x^3+1 is invertible?

If f(x) is invertible, then f(a) = f(b) implies a = b

f(-3) = 81 - 81 + 1

= 1

f(0) = 1

But 0 is not equal to -3

Hence f(x) not invertible

no , the derivative has a sign change at x = - 9 / 4

Well,

lim (x ---> -oo) f(x) = +oo
and
lim (x ---> +oo) f(x) = +oo
and
f(0) = 1
therefore, as the function is continuous
there exists x1 € (-oo, 0) so that : f(x1) = 10 (could be any other number greater than 1 !! )
and
there exists x2 € (0, +oo) so that : f(x2) = 10
conclusion :
10 has at least two antecedents ==> f is not invertible

hope it' ll help !!