Advertisements
Advertisements
Question
Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-1)/3 .`
Advertisements
Solution
Injectivity of f :
Let x and y be two elements of domain` (R^+)`, such that
f(x)=f(y)
⇒ 9x2+6x−5=9y2+ 6y − 5
⇒ 9x2+6x=9y2+6y
⇒ x = y (As, x, y ∈ `R^+`)
So, f is one-one.
Surjectivity of f:
Let y is in the co domain (Q) such that f(x) = y
⇒ 9x2 + 6x - 5 = y
⇒ 9x2 +6x = y + 5
⇒ 9x2 + 6x +1 = y +6 (Adding 1 on both sides )
⇒ (3x +1)2 = y + 6
⇒ `3x +1 = sqrt(y + 6)`
⇒ `3x = sqrt (y + 6) -1`
⇒ `x = (sqrt (y + 6)-1)/3 in R^+` (domain)
f is onto.
So, f is a bijection and hence, it is invertible.
Finding `f^-1`
Let f−1(x) = y ...(1)
⇒ x = f (y)
⇒ x = 9y2+ 6y − 5
⇒ x + 5 = 9y2+6y
⇒ x + 6= 9y2+ 6y + 1 (adding 1 on both sides)
⇒ x + 6 = ( 3y + 1 )2
⇒3y+1=`sqrt(x +6)`
⇒ `3y = sqrt (x +6) -1`
⇒ `y = (sqrt (x+6)-1)/3`
`So, f^-1 (x) (sqrt (x-6)-1)/3 ` [from (1)]
APPEARS IN
RELATED QUESTIONS
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. Show that f is one-one.
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
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) = sin−1 x, g(x) = x2
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
If f(x) = |x|, prove that fof = f.
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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 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?
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
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]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
Let
\[f : R - \left\{ n \right\} \to R\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is
The function f : R → R given by f(x) = x3 – 1 is ____________.
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 ____________.
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Raji wants to know the number of functions from A to B. How many number of functions are possible?
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 ____________.
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 ____________.
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The given function f : R → R is not ‘onto’ function. Give reason.
