Let A-1-0-1 and f-x- x2- x - A- Show that f- A - A is neither one-one nor onto-

A = minus;1, 0, 1 and f = (x, x2) : x isin; A Given, f(x) = x2 Injectivity: f(1) = 12=1 and f( 1)=( 1)2=1 #8658;1 and 1 have the same images. So, f ...

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Standard XII

Mathematics

Geometrical Explanation of Rolle's Theorem

Let A=-1,0,1 ...

Question

Let A = {−1, 0, 1} and f = {(x, x²) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

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Solution

A = {−1, 0, 1} and f = {(x, x²) : x ∈ A}
Given, f(x) = x²

Injectivity:
f(1) = 1²=1 and
f(-1)=(-1)²=1

$\Rightarrow$ 1 and -1 have the same images.
So, f is not one-one.

Surjectivity:
Co-domain of f = {-1, 0, 1}

f(1) = 1² = 1,
f(-1) = (-1)² = 1 and
f(0) = 0
$\Rightarrow$ Range of f = {0, 1}
So, both are not same.
Hence, f is not onto.

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Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.

Let A = {−1, 0, 1} and f = {(x, x²) : x ∈ A}. Show that f : A → A is neither one-one nor onto.