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
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Show that the function f : R → R given by f(x) = x3 is injective.
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}
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
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.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
Find fog and gof if : f (x) = x2 g(x) = cos x .
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
If f : C → C is defined by f(x) = x4, write f−1 (1).
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 be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
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. State whether f is one-one or not.
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
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
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.
Let A be a finite set. Then, each injective function from A into itself is not surjective.
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, 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: N → N be defined by f(x) = x2 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 ____________.
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
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.
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
`x^(log_5x) > 5` implies ______.
Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.
