Class XII Chapter 1 – Relations and Functions: Functions and One-One Functions

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What is the purpose of defining the function $g(y) = \frac{y-7}{10}$ in the given text?

To show that $f$ is onto
To simplify the expression for $f$
To show that $f$ is one-to-one
To find the inverse of $f$ (correct)

What is the relationship between the functions $g \circ f$ and $f \circ g$?

$g \circ f = I_R$ and $f \circ g = I_R$ (correct)
$g \circ f = I_{\mathbb{R}^2}$ and $f \circ g = I_{\mathbb{R}^2}$
$g \circ f = I_W$ and $f \circ g = I_W$
$g \circ f = I_\mathbb{R}$ and $f \circ g = I_\mathbb{R}$

Why is the function $f(n) = n - 1$ for odd $n$ and $f(n) = n + 1$ for even $n$ invertible?

Because $f$ is a linear function
Because $f$ is a bijective function
Because $f$ is an onto function
Because $f$ is a one-to-one function (correct)

What is the inverse function of $f(n) = n - 1$ for odd $n$ and $f(n) = n + 1$ for even $n$?

$f^{-1}(n) = n - 1$ for odd $n$ and $f^{-1}(n) = n + 1$ for even $n$ (D) Signup and view all the answers

Which of the following is a characteristic of an invertible function?

All of the above (D) Signup and view all the answers

Why is the function $f(x) = 10x + 7$ invertible?

Because $f$ is both one-to-one and onto (A) Signup and view all the answers

Which of the following statements is true about the function f(x) = { x^2 for all x ∈ R , if x is even 2 }?

The function is both one-one and onto (A) Signup and view all the answers

For the function f: A × B → B × A defined as f(a, b) = (b, a), which of the following is true?

The function is both one-one and onto (C) Signup and view all the answers

In the given function f: N → N defined as f(n) = n^2 for all n ∈ N, what can be said about its bijectiveness?

The function is one-one but not onto (A) Signup and view all the answers

For the function f: R × R → R × R where f(x1, x2) = (x2, x1), what can be said about its bijectiveness?

The function is bijective (A) Signup and view all the answers

Consider the function f(x) = 1 + x^2 for all x ∈ R. Is this function one-one?

No, the function is not one-one (C) Signup and view all the answers

For the given bijective function f: A → B, if f(a) = b and f(b) = a, what can be concluded about the relationship between sets A and B?

|A| = |B| (B) Signup and view all the answers

What does it mean for a function $f: A \rightarrow B$ to be one-one?

For every $x, y \in A$, if $x \neq y$, then $f(x) \neq f(y)$ (C) Signup and view all the answers

In the given example, why is the function $f: \mathbb{N} \rightarrow \mathbb{N}$ defined by $f(n) = \frac{n+1}{2}$ not one-one?

For $n=1$ and $n=2$, $f(1) = f(2) = 1$, violating the one-one condition (D) Signup and view all the answers

Which of the following functions is one-one?

$k: \mathbb{R} \rightarrow \mathbb{R}$ defined by $k(x) = x + 1$ (B) Signup and view all the answers

If $f: A \rightarrow B$ is one-one and $g: B \rightarrow C$ is one-one, is the composite function $g \circ f: A \rightarrow C$ necessarily one-one?

Yes, because the composition of two one-one functions is always one-one (A) Signup and view all the answers

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x) = 3x + 2$. Which of the following statements is true?

$f$ is both one-one and onto (C) Signup and view all the answers

Let $A = {1, 2, 3, 4}$ and $B = {5, 6, 7, 8}$. How many one-one functions are there from $A$ to $B$?

4! (C) Signup and view all the answers