मराठी

If F ( X ) = X 3 − 1 X 3 , Show that F ( X ) + F ( 1 X ) = 0 . - Mathematics

Advertisements
Advertisements

प्रश्न

If  \[f\left( x \right) = x^3 - \frac{1}{x^3}\] , show that

\[f\left( x \right) + f\left( \frac{1}{x} \right) = 0 .\]
 

 

उत्तर

Given:

\[f\left( x \right) = x^3 - \frac{1}{x^3}\]    ...(i)
Thus,
\[f\left( \frac{1}{x} \right) = \left( \frac{1}{x} \right)^3 - \frac{1}{\left( \frac{1}{x} \right)^3}\] \[= \frac{1}{x^3} - \frac{1}{\frac{1}{x^3}}\]
\[\therefore f\left( \frac{1}{x} \right) = \frac{1}{x^3} - x^3\]  ...(ii) 
\[f\left( x \right) + f\left( \frac{1}{x} \right) = \left( x^3 - \frac{1}{x^3} \right) + \left( \frac{1}{x^3} - x^3 \right)\]
\[= x^3 - \frac{1}{x^3} + \frac{1}{x^3} - x^3 = 0\] 
Hence,
\[f\left( x \right) + f\left( \frac{1}{x} \right) = 0\]
shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 3: Functions - Exercise 3.2 [पृष्ठ ११]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 11
पाठ 3 Functions
Exercise 3.2 | Q 7 | पृष्ठ ११

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

  1. {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}
  2. {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}
  3. {(1, 3), (1, 5), (2, 5)}

Let X = {1, 2, 3, 4} and Y = {1, 5, 9, 11, 15, 16}
Determine which of the set are functions from X to Y.

(c) f3 = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

 

 


Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(iv) \[\frac{f}{g}\]

 

Let f and g be two real functions defined by \[f\left( x \right) = \sqrt{x + 1}\] and \[g\left( x \right) = \sqrt{9 - x^2}\] . Then, describe function: 

(viii) \[\frac{5}{8}\]

 

Write the domain and range of function f(x) given by \[f\left( x \right) = \sqrt{\left[ x \right] - x}\] .

 

Find the set of values of x for which the functions f(x) = 3x2 − 1 and g(x) = 3 + x are equal.


Which one of the following is not a function?


Let f : R → R be defined by f(x) = 2x + |x|. Then f(2x) + f(−x) − f(x) =


The range of the function  \[f\left( x \right) = \frac{x^2 - x}{x^2 + 2x}\]  is 

 

The range of  \[f\left( x \right) = \frac{1}{1 - 2\cos x}\] is 

 


If f(m) = m2 − 3m + 1, find f(−3)


If f(x) =` (2x−1)/ (5x−2) , x ≠ 2/5` Verify whether (fof) (x) = x


Check if the relation given by the equation represents y as function of x:

x2 − y = 25


If f(m) = m2 − 3m + 1, find f(0)


Find x, if g(x) = 0 where g(x) = `(5x - 6)/7`


Find x, if g(x) = 0 where g(x) = x3 − 2x2 − 5x + 6


Find the domain and range of the following function.

f(x) = `sqrt((x - 3)/(7 - x))`


lf f(x) = 3(4x+1), find f(– 3)


Express the following exponential equation in logarithmic form

e2 = 7.3890


Express the following logarithmic equation in exponential form

`log_5  1/25` = – 2


Select the correct answer from given alternatives

If f(x) = 2x2 + bx + c and f(0) = 3 and f(2) = 1, then f(1) is equal to


Select the correct answer from given alternatives

The domain of `1/([x] - x)` where [x] is greatest integer function is


Answer the following:

A function f is defined as : f(x) = 5 – x for 0 ≤ x ≤ 4. Find the value of x such that f(x) = 5


Answer the following:

If a2 + b2 = 7ab, show that, `log(("a" + "b")/3) = 1/2 log "a" + 1/2 log "b"`


Answer the following:

Without using log tables, prove that `2/5 < log_10 3 < 1/2`


Answer the following:

Find the range of the following function.

f(x) = 1 + 2x + 4x 


Let f = {(x, y) | x, y ∈ N and y = 2x} be a relation on N. Find the domain, co-domain and range. Is this relation a function?


A graph representing the function f(x) is given in it is clear that f(9) = 2

Describe the following Range


An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal square from the corner and turning up the side as shown. Express the volume V of the box as a function of x


The data in the adjacent table depicts the length of a person's forehand and their corresponding height. Based on this data, a student finds a relationship between the height (y) and the forehand length (x) as y = ax + b, where a, b are constant.

Length ‘x’ of
forehand (in cm)
Height 'y' 
(in inches)
35 56
45 65
50 69.5
55 74

Find the height of a person whose forehand length is 40 cm


Domain of function f(x) = cos–1 6x is ______.


Find the range of the following functions given by f(x) = `3/(2 - x^2)`


Find the domain and range of the function f(x) = `1/sqrt(x - 5)`


The domain of the function f(x) = `sin^-1((|x| + 5)/(x^2 + 1))` is (–∞, –a] ≈ [a, ∞). Then a is equal to ______.


Let f(θ) = sin θ (sin θ + sin 3θ) then ______.


Range of the function f(x) = `x/(1 + x^2)` is ______.


The domain of f(x) = `sin^-1 [log_2(x/2)]` is ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×