Solution to 15.Determine whether f: R to R, defined as f (x) = −3x + 4 is … - Sikademy
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15.Determine whether f: R to R, defined as f (x) = −3x + 4 is a bijection. Is f invertible, and if it is, what is its inverse? 16.Find the inverse of f (x) = 𝑥+1 𝑥+2 , on a suitable subset of R.

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15.

Let x_1, x_2 \in \R and f(x) = -3x+4.

If f(x_1)=f(x_2)=>-3x_1+4=-3x_2+4

=>-3x_1=-3x_2=>x_1=x_2.

If x_1\not=x_2=>f(x_1)\not=f(x_2).

Then

f(x_1)=f(x_2)<=>x_1=x_2, x_1, x_2\in \R

The function f(x)=-3x+4 is one-to-one.


\forall y\in\R, y=-3x+4, \exist x=-\dfrac{1}{3}y+\dfrac{4}{3}, x\in \R

The function f(x)=-3x+4 is onto.


The function f(x)=-3x+4 is one-to-one and is onto.

Therefore the function f(x)=-3x+4 is a bijection. Therefore the function f(x)=-3x+4 is invertibele


f(x)=-3x+4

Replace f(x) with y


y=-3x+4

Switch x and y


x=-3y+4

Solve for y


y=-\dfrac{1}{3}x+\dfrac{4}{3}

Replace y with f^{-1}(x)


f^{-1}(x)=-\dfrac{1}{3}x+\dfrac{4}{3}

16.


f(x)=\dfrac{x+1}{x+2}

Domain: (-\infin, -2)\cup (-2, \infin)

Replace f(x) with y


y=\dfrac{x+1}{x+2}

Switch x and y


x=\dfrac{y+1}{y+2}

Solve for y


xy+2x=y+1


y=-\dfrac{2x-1}{x-1}

Replace y with f^{-1}(x)


f^{-1}(x)=-\dfrac{2x-1}{x-1}

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