Which of these functions is (are) always increasing?

Question

Which of these functions is (are) always increasing?
(I) f(x) = -x2 + 2x + 1
(II) f(x) = 2x - 1000
(III) f(x) = |x - 4| + 255
(IV) f(x) = -e-x

 

A) (I) and (II) only
B) (II) only
C) (II) and (IV) only
D) (I) and (III) only
E) (III) and (IV) only 

Answer 1

(I) is a quadratic function, which has the form of a parabola.  A parabola has one side that increases and one side that decreases.  NO

 

(II) is a linear function with a slope of 2.  A positive slope means that it is always increasing.  YES

 

(III) takes the an absolute value of a linear expression, x-4.  These are always shaped like a V or an upside-down V, which means that like the parabola, they have one side that increases and one side that decreases.  NO

 

(IV) I assume you mean -e^-x.  e^x, the basic exponential function always increases, faster and faster as x gets larger.  e^-x is the reciprocal of e^x, so it must always decrease.  So -e^-x, the negative of  e^-x, must always increase.  YES

 

So the answer is C, (II) and (IV) only.