Advertisements
Advertisements
प्रश्न
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 8x3 and g(x) = x1/3.
Advertisements
उत्तर
Given, f : R → R and g : R → R
So, gof : R → R and fog : R → R
f(x) = 8x3 and g(x) = x1/3
(gof) (x)
= g (f (x))
= g (8x3)
=`(8x^3)^(1/3)`
= `[(2x)^3]^(1/3)`
= 2x
(fog) (x)
= f (g (x))
=` f (x^(1/3))`
=` 8 (x^(1/3))^3`
= 8x
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = `((x - 2)/(x - 3))`. Is f one-one and onto? Justify your answer.
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 1)}
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
If f(x) = |x|, prove that fof = f.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.
Write the domain of the real function
`f (x) = sqrtx - [x] .`
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
k = {(1,4), (2, 5)}
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
The function f: R → R defined as f(x) = x3 is:
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Based on the given information, f is best defined as:
Let f: R → R defined by f(x) = 3x. Choose the correct answer
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
Let f: R→R be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ R. If I1 = `int_0^8f(x)dx` and I2 = `int_(-1)^3f(x)dx`, then the value of I1 + 2I2 is equal to ______.
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
