Question

If f:R is defined by f(x)=5x-7.show that f is bijective & find the inverse function.

Answer 1

SOLUTION

GIVEN

A function f : R → R is defined by f(x)= 5x - 7

TO EVALUATE

1. f is bijection

$\sf{2. \: \: To \: find \: \: {f}^{ - 1} \: \: }$

EVALUATION

CHECKING FOR ONE TO ONE

$\sf{Let \: x, y \in \mathbb{R} \: \: such \: that \: f(x) = f(y) }$

Now f(x) = f(y) gives

$\sf{5x - 7 = 5y - 7\: }$

$\implies \sf{5x = 5y}$

$\implies \sf{x = y}$

Hence f is one to one

CHECKING FOR ONTO

$\sf{Let \: y \in \mathbb{R} }$

$\sf{If \: possible \: there \: exists \: x \in \mathbb{Q}} \: such \: that \: f(x) = y$

Which gives $\sf{ 5x - 7 = y\: }$

$\implies \displaystyle \sf{ x = \frac{y +7}{5} \: }$

$\displaystyle \sf{ As \: \: y \in \mathbb{R} \: \: \: so \: \: \frac{y +7}{5} \:\in \mathbb{R} }$

Since y is arbitrary

So f is onto

Hence f is bijection

$\sf{Hence \: \: {f}^{ - 1} \: \: exists}$

DETERMINATION OF INVERSE OF THE FUNCTION

$\displaystyle \sf{ let \: \: {f}^{ - 1}(x) = y \: }$

$\implies \displaystyle \sf{ x =f(y) \: }$

$\implies \displaystyle \sf{ x = 5y - 7\: }$

$\implies \displaystyle \sf{ y = \frac{x +7}{5} \: }$

So$\displaystyle \sf{ {f}^{ - 1} (x)= \frac{x +7}{5} \: }$

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