f-x-f-1-1-x-3-1-3-x-3-is-given-find-the-value-of-f-1-

Question Number 177530 by mnjuly1970 last updated on 06/Oct/22

f (x) + f (\frac{1}{\sqrt[3]{1 - x^{3}}}) = x^{3}

i s g i v e n . f i n d t h e v a l u e o f :

f (- 1) = ?

Answered by floor(10²Eta[1]) last updated on 06/Oct/22

f (x) + f (\frac{1}{\sqrt[3]{1 - x^{3}}}) = x^{3} (1)

x \to \frac{1}{\sqrt[3]{1 - x^{3}}}

f (\frac{1}{\sqrt[3]{1 - x^{3}}}) + f (\frac{\sqrt[3]{x^{3} - 1}}{x}) = \frac{1}{1 - x^{3}} (2)

x \to \frac{1}{\sqrt[3]{1 - x^{3}}}

f (\frac{\sqrt[3]{x^{3} - 1}}{x}) + f (x) = \frac{x^{3} - 1}{x^{3}} (3)

(3) - (2) :

f (x) - f (\frac{1}{\sqrt[3]{1 - x^{3}}}) = \frac{x^{3} - 1}{x^{3}} - \frac{1}{1 - x^{3}} (4)

(1) + (4) :

f (x) = \frac{1}{2} (\frac{x^{3} - 1}{x^{3}} - \frac{1}{1 - x^{3}} + x^{3})

\Rightarrow f (- 1) = \frac{1}{4}

Commented by mnjuly1970 last updated on 07/Oct/22

t h a n k s s i r

Answered by mr W last updated on 06/Oct/22

f (- 1) + f (\frac{1}{\sqrt[3]{2}}) = - 1 \dots (i)

f (\frac{1}{\sqrt[3]{2}}) + f (\sqrt[3]{2}) = \frac{1}{2} \dots (i i)

f (\sqrt[3]{2}) + f (- 1) = 2 \dots (i i i)

(i) + (i i i) - (i i) :

2 f (- 1) = 2 - 1 - \frac{1}{2} = \frac{1}{2}

\Rightarrow f (- 1) = \frac{1}{4}

Commented by mnjuly1970 last updated on 07/Oct/22

t h a n k s a l o t s i r W

Answered by a.lgnaoui last updated on 06/Oct/22

p o u r x =^{3} \sqrt{2}

f (^{3} \sqrt{2}) + f \frac{1}{^{3} \sqrt{1 - 2}} = 2 (d a p r e s l e x p r e s s i i n)

d o n c f (^{3} \sqrt{2}) + f (- 1) = 2 (1)

p o u r x = - 1

f (- 1) + f (\frac{1}{^{3} \sqrt{1 - {(- 1)}^{3}}}) = f (- 1) + f (\frac{1}{^{3} \sqrt{2}}) = - 1 (2)

(1) e t (2) \Rightarrow 2 - f (^{3} \sqrt{2}) = - 1 - f (\frac{1}{^{3} \sqrt{2}})

s o i t f (^{3} \sqrt{2}) - f (\frac{1}{^{3} \sqrt{2}}) = 3 (3)

C a l c u l d e f (\frac{1}{^{3} \sqrt{2}}) ?

f (\frac{1}{^{3} \sqrt{2}}) + f (\frac{1}{^{3} \sqrt{1 - \frac{1}{2}}}) = \frac{1}{2}

f (\frac{1}{^{3} \sqrt{2}}) + f (^{3} \sqrt{2}) = \frac{1}{2} (4)

(3) e t (4) \Rightarrow f (\frac{1}{^{3} \sqrt{2}}) = f (^{3} \sqrt{2}) - 3 = \frac{1}{2} - f (- 1)

(3) e t (4) \Rightarrow {\begin{cases} f (^{3} \sqrt{2}) - f (\frac{1}{^{3} \sqrt{2}}) = 3 \\ f (^{3} \sqrt{2}) + f (\frac{1}{^{3} \sqrt{2}}) = \frac{1}{2} \end{cases}

2 f (^{3} \sqrt{2}) = \frac{7}{2} f (\sqrt{2}) = \frac{7}{4}

f i n a l e m e n t e n [r e m p l a c a n t d a n s (1)

f (\sqrt{2}) + f (- 1) = 2

f (- 1) = 2 - f (^{3} \sqrt{2}) = 2 - \frac{7}{4}

f (- 1) = \frac{1}{4}