Advertisements
Advertisements
प्रश्न
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Advertisements
उत्तर
A =R − {2}, B = R − {1}
f: A → B is defined as `"f"("x") = ("x"-1)/("x"-2)`.
Let x, y ∈ A such that f (x) = f (y).
⇒ `("x"-1)/("x"-2) = ("y"-1)/("y"-2)`
⇒ ( x -1) (y - 2) = (x - 2) (y - 1)
⇒ xy - 2x - y + 2 = xy - x - 2y + 2
⇒ -2x - y = -x - 2y
⇒ 2x - x = 2y - y
⇒ x = y
∴ f is one-one.
Let y ∈B = R − {1}. Then, y ≠ 1.
The function f is onto if there exists x ∈A such that f(x) = y.
Now,
f (x) = y
⇒`("x"-1)/("x"-2) = "y"`
⇒ x - 1 = y (x - 2)
⇒ x (1 - y) = 1 - 2y
⇒ `"x" = (1-2"y")/(1-"y")∈ "A"` .........[y ≠ 1]
Thus, for any y ∈ B, there exists `"x" = (1-2"y")/(1-"y")` ∈ A such that
`"f"((1-2"y")/(1-"y")) =((1-2"y")/(1-"y")-1)/((1-2"y")/(1-"y") - 2) = (1-2"y"-1+"y")/(1-2"y"-2+2"y") = (-"y")/-1 = "y"`
Therefore, f is onto.
Hence, function f is one-one and onto.
`"f"^-1("x") = (1-2"x")/(1-"x")`
APPEARS IN
संबंधित प्रश्न
Let f : R → R be defined as f(x) = x4. Choose the correct answer.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Find fog and gof if : f (x) = x2 g(x) = cos x .
State with reason whether the following functions have inverse :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
Which of the following graphs represents a one-one function?
Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
Which of the following functions form Z to itself are bijections?
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Write about strlen() function.
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- The function f: R → R defined by f(x) = x − 4 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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
A function f: x → y is said to be one – one (or injective) if:
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
