Advertisements
Advertisements
प्रश्न
If f is a real function satisfying \[f\left( x + \frac{1}{x} \right) = x^2 + \frac{1}{x^2}\]
for all x ∈ R − {0}, then write the expression for f(x).
उत्तर
Given:
\[f\left( x + \frac{1}{x} \right) = x^2 + \frac{1}{x^2}\]
\[= x^2 + \frac{1}{x^2} + 2 - 2\]
f (x) = x2 - 2 , where | x | ≥ 2.
APPEARS IN
संबंधित प्रश्न
Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.
If \[f\left( x \right) = \frac{x - 1}{x + 1}\] , then show that
(i) \[f\left( \frac{1}{x} \right) = - f\left( x \right)\]
(ii) \[f\left( - \frac{1}{x} \right) = - \frac{1}{f\left( x \right)}\]
If f(x) = loge (1 − x) and g(x) = [x], then determine function:
(iii) \[\frac{f}{g}\]
Let f : [0, ∞) → R and g : R → R be defined by \[f\left( x \right) = \sqrt{x}\] and g(x) = x. Find f + g, f − g, fg and \[\frac{f}{g}\] .
If f(x) = cos [π2]x + cos [−π2] x, where [x] denotes the greatest integer less than or equal to x, then write the value of f(π).
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] . Then write the value of α satisfying f(f(x)) = x for all x ≠ −1.
Let A = {1, 2, 3} and B = {2, 3, 4}. Then which of the following is a function from A to B?
If A = {1, 2, 3} and B = {x, y}, then the number of functions that can be defined from A into B is
If x ≠ 1 and \[f\left( x \right) = \frac{x + 1}{x - 1}\] is a real function, then f(f(f(2))) is
If f : R → R and g : R → R are defined by f(x) = 2x + 3 and g(x) = x2 + 7, then the values of x such that g(f(x)) = 8 are
If f : [−2, 2] → R is defined by \[f\left( x \right) = \begin{cases}- 1, & \text{ for } - 2 \leq x \leq 0 \\ x - 1, & \text{ for } 0 \leq x \leq 2\end{cases}\] , then
{x ∈ [−2, 2] : x ≤ 0 and f (|x|) = x} =
If \[3f\left( x \right) + 5f\left( \frac{1}{x} \right) = \frac{1}{x} - 3\] for all non-zero x, then f(x) =
The domain of the function
If f(m) = m2 − 3m + 1, find `f(1/2)`
Check if the following relation is a function.
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 2), (2, −1), (3, 1), (4, 3)}
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 3), (4, 1), (2, 2)}
Which sets of ordered pairs represent functions from A = {1, 2, 3, 4} to B = {−1, 0, 1, 2, 3}? Justify.
{(1, 1), (2, 1), (3, 1), (4, 1)}
Check if the relation given by the equation represents y as function of x:
x2 − y = 25
Find the domain and range of the following function.
g(x) = `(x + 4)/(x - 2)`
Find the domain and range of the follwoing function.
h(x) = `sqrt(x + 5)/(5 + x)`
Find the domain and range of the following function.
f(x) = `sqrt((x - 2)(5 - x)`
Express the area A of a square as a function of its side s
Express the area A of a square as a function of its perimeter P
Let f be a subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify?
Express the following logarithmic equation in exponential form
ln e = 1
Write the following expression as sum or difference of logarithm
`log (sqrt(x) root(3)(y))`
Write the following expression as sum or difference of logarithm
In `(("a"^3 ("a" - 2)^2)/sqrt("b"^2 + 5))`
Answer the following:
Find whether the following function is one-one
f : R → R defined by f(x) = x2 + 5
Answer the following:
A function f is defined as f(x) = 4x + 5, for – 4 ≤ x < 0. Find the values of f(–1), f(–2), f(0), if they exist
Answer the following:
Let f : R → R be given by f(x) = x + 5 for all x ∈ R. Draw its graph
If f(x) = `x^3 - 1/x^3`, then `f(x) + f(1/x)` is equal to ______.
Find the range of the following functions given by f(x) = `3/(2 - x^2)`
If f(x) = `(x - 1)/(x + 1)`, then show that `f(1/x)` = – f(x)
If f(x) = y = `(ax - b)/(cx - a)`, then prove that f(y) = x.
The range of the function y = `1/(2 - sin3x)` is ______.
The expression \[\begin{array}{cc}\log_p\log_p\sqrt[p]{\sqrt[p]{\sqrt[p]{\text{...........}\sqrt[p]{p}}}}\\
\phantom{...........}\ce{\underset{n radical signs}{\underline{\uparrow\phantom{........}\uparrow}}}
\end{array}\]where p ≥ 2, p ∈ N; ∈ N when simplified is ______.
