range of x^3 — step-by-step solution

Let's find the real values taken by

x^{3}

using its variation table

⇥Let's study the variations of

f (x)

: first find where its derivative

f^{'} (x) = 3 x^{2}

is zero

⇥Let's solve in

ℝ

3 x^{2} = 0

⇥We simplify the expression by

3

⇥⇥Let's solve in

ℝ

x^{2} = 0

⇥⇥

x^{2} = 0

is equivalent to:

x = 0

⇥⇥Solution obtained:

x = 0

⇥

f^{'} (x)

is zero at

x = 0

⇥Built from usual functions,

f^{'} (x)

is continuous where it is defined: on each interval where it is never zero, it keeps the sign of its value at any point

⇥

f^{'} (-1) = 3

> 0:

f

is increasing on

(-∞, 0)

⇥

f^{'} (1) = 3

> 0:

f

is increasing on

(0, +∞)

⇥Let's calculate the values of

f

at the bounds of the table, or its limits at the infinite or excluded bounds

⇥As

x

tends to

-∞

f (x)

tends to

-∞

⇥

f (0) = 0^{3} = 0

⇥As

x

tends to

+∞

f (x)

tends to

+∞

⇥Let's draw up the variation table:

⇥

⇥So

f

is increasing on

ℝ

ℝ

f

is continuous: it takes every value of

ℝ

▶Range:

ℝ