Advertisements
Advertisements
प्रश्न
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
उत्तर
Here the function f : R → R is defined as f (x) = 4x + 5 = y (say). Then
4x = y – 5 or x = `(y - 5)/4`.
This leads to a function g: R → R defined as
g(y) = `(y - 5)/4`.
Therefore, (gof) (x) = g(f(x) = g(4x + 5)
= `(4x + 5 - 5)/4`
= x
or
gof = IR
Similarly (fog) (y) = f(g(y))
= `f((y - 5)/4)`
= `4((y - 5)/4) + 5`
= y
or
fog = IR
Hence f is invertible and f-1 = g which is given by `f^-1 (x) = (x - 5)/4`
APPEARS IN
संबंधित प्रश्न
Find the number of all onto functions from the set {1, 2, 3, …, n} to itself.
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
If f(x) = |x|, prove that fof = f.
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)}
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
If f : R → R is given by f(x) = x3, write f−1 (1).
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
Which of the following functions form Z to itself are bijections?
Which of the following functions from
to itself are bijections?
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
The function
The distinct linear functions that map [−1, 1] onto [0, 2] are
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
Let g(x) = x2 – 4x – 5, then ____________.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)-1 (x) is equal to ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
A function f: x → y is said to be one – one (or injective) if:
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
