Suppose f1 and f2 are non-zero one-one functions from R to R- Is f1f2 necessarily one-one- Justify your answer- Here- f1f2- R - R is given by f1f2 x-f 1 x f 2 x for all x - R

We know that f1: R rarr; R, given by f1(x)=x3 and f2(x)=x are one one. Injectivity of f1: Let x and y be two elements in the domain R, such that f1x=f2y #8 ...

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Standard XII

Mathematics

Conjugate of a Matrix

Suppose f1 an...

Question

Suppose f₁ and f₂ are non-zero one-one functions from R to R. Is $\frac{f_{1}}{f_{2}}$ necessarily one-one? Justify your answer. Here, $\frac{f_{1}}{f_{2}} : R \to R$ is given by $(\frac{f_{1}}{f_{2}}) (x) = \frac{f_{1} (x)}{f_{2} (x)}$ for all $x \in R$ .

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Solution

We know that f₁: R → R, given by f₁(x)=x³ and f₂(x)=x are one-one.
Injectivity of f₁:
Let x and y be two elements in the domain R, such that
$f_{1} (x) = f_{2} (y) \Rightarrow x^{3} = y \Rightarrow x = \sqrt[3]{y} \in R$ Let f1(x)=f1(y)x=y
So, f₁is one-one.

Injectivity of f₂:
Let x and y be two elements in the domain R, such that
$f_{2} (x) = f_{2} (y) \Rightarrow x = y ⇒ x \in R .$ Let f2(x)=f2(y)−x=−yx=y
So, f₂ is one-one.

Proving $\frac{f_{1}}{f_{2}}$ is not one-one:
Given that $\frac{f_{1}}{f_{2}} (x) = \frac{f_{1} (x)}{f_{2} (x)} = \frac{x^{3}}{x} = x^{2}$
Let x and y be two elements in the domain R, such that

$\frac{f_{1}}{f_{2}} (x) = \frac{f_{1}}{f_{2}} (y) \Rightarrow x^{2} = y^{2} \Rightarrow x = \pm y$
So, $\frac{f_{1}}{f_{2}}$ is not one-one.

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Suppose f1 and f2 are non-zero one-one functions from R to R. Is f1f2 necessarily one-one? Justify your answer. Here, f1f2:R→R is given by f1f2 x=f1 xf2 x for all x∈R.

Suppose f₁ and f₂ are non-zero one-one functions from R to R. Is $\frac{f_{1}}{f_{2}}$ necessarily one-one? Justify your answer. Here, $\frac{f_{1}}{f_{2}} : R \to R$ is given by $(\frac{f_{1}}{f_{2}}) (x) = \frac{f_{1} (x)}{f_{2} (x)}$ for all $x \in R$ .