Advertisements
Advertisements
प्रश्न
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
Advertisements
उत्तर
Given, `f(x) = (2x - 7)/4`
For one-one
Let x, x2 ∈ R
f(x1) = f(x2)
`(2x_1 - 7)/4 = (2x_2 - 7)/4`
`\implies` 2x1 – 7 = 2x2 – 7
`\implies` 2x1 = 2x2
`\implies` x1 = x2
So, f(x) is a one-to-one function.
For onto
Put `y = (2x - 7)/4`
`\implies` 4y = 2x – 7
`\implies` 4y + 7 = 2x
`\implies` `x = (4y + 7)/2` ∀ y ∈ R ∃ a unique x ∈ R
Therefore, f(x) is onto.
APPEARS IN
संबंधित प्रश्न
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
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.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
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) = x3 − x
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f (x) = x2 g(x) = cos x .
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
Which of the following graphs represents a one-one function?
If f : R → R is given by f(x) = x3, write f−1 (1).
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, 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 ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
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 ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
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 ______.
Find the domain of sin–1 (x2 – 4).
