Advertisements
Advertisements
Question
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
Advertisements
Solution
`f (x) = sqrt(1-x)`
For domain, 1-x≥0
⇒ x≤1
⇒ domain of f = (−∞, 1]
⇒ f : (−∞, 1] → (0,∞)
g(x) = loge x
Clearly, g : (0, ∞) → R
Computation of fog :
Clearly, the range of g is not a subset of the domain of f.
So,we need to compute the domain of fog.
⇒ Domain (fog) = {x : x ∈ Domain (g) and g(x) ∈ Domain of f}
⇒ Domain (fog) = {x: x ∈ (0, ∞) and loge x ∈ (−∞, 1]}
⇒ Domain (fog) = { x: x ∈ (0, ∞) and x ∈ (0, e] }
⇒ Domain (fog)= {x : x ∈ (0, e]}
⇒ Domain (fog )= (0, e]
⇒ fog : (0, e) → R
So, (fog) (x) = f (g (x))
= f (loge x)
= `sqrt( 1-log_e x)`
Computation of gof:Clearly, the range of f is a subset of the domain of g.
⇒ gof : (−∞,1] → R
⇒ (gof) (x) = g (f (x))
= `g (sqrt(1-x))`
= `log_e sqrt (1 - x)`
= `log_e (1 - x)^(1/2)`
= `1/2 log_e (1-x)`
APPEARS IN
RELATED QUESTIONS
Show that the modulus function f : R → R, given by f(x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.
Give an example of a function which is neither one-one nor onto ?
Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
If f(x) = |x|, prove that fof = f.
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.
If f : R → R is defined by f(x) = x2, write f−1 (25)
Let \[f : \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \to R\] be a function defined by f(x) = cos [x]. Write range (f).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
The function
The inverse of the function
\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by
\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Let A be a finite set. Then, each injective function from A into itself is not surjective.
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 defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
Let f(1, 3) `rightarrow` R be a function defined by f(x) = `(x[x])/(1 + x^2)`, where [x] denotes the greatest integer ≤ x, Then the range of f is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
