Show that if f1 and f2 are one-one maps from R to R- then the product f 1 - f 2- R - R defined by f 1 - f 2 x-f 1 - f2 x need not be one-one-

We know that f1: R rarr; R, given by f1(x) = x, and f2(x) = x are one one. Proving f1 is one one: Let x and y be two elements in the domain R, such that f1(x ...

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Standard XII

Mathematics

Common Ratio

Show that if ...

Question

Show that if f₁ and f₂ are one-one maps from R to R, then the product $f_{1} \times f_{2} : R \to R$ defined by $(f_{1} \times f_{2}) (x) = f_{1} (x) f_{2} (x)$ need not be one-one.

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Solution

We know that f₁: R → R, given by f₁(x) = x, and f₂(x) = x are one-one.
Proving f₁is one-one:
Let x and y be two elements in the domain R, such that
f₁(x) = f₁(y)
$\Rightarrow$ x = yet f1(x)=f1(y)x=y
So, f₁is one-one.

Proving f₂is one-one:
Let x and y be two elements in the domain R, such that
f₂(x) = f₂(y)
$\Rightarrow$ x = yet f1(x)=f1(y)x=y
So, f₂is one-one.

Proving $f_{1} \times f_{2}$ is not one-one:
Given:
$(f_{1} \times f_{2}) (x) = f_{1} (x) \times f_{2} (x) = x \times x = x^{2} Let x and y be two elements in the domain R, such that (f_{1} \times f_{2}) (x) = (f_{1} \times f_{2}) (y) \Rightarrow x^{2} = y^{2} \Rightarrow x = \pm y So, (f_{1} \times f_{2}) is not one-one.$ f1×f2

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Show that if f1 and f2 are one-one maps from R to R, then the product f1×f2 : R→R defined by f1×f2 x=f1 x f2 x need not be one-one.

Show that if f₁ and f₂ are one-one maps from R to R, then the product $f_{1} \times f_{2} : R \to R$ defined by $(f_{1} \times f_{2}) (x) = f_{1} (x) f_{2} (x)$ need not be one-one.