Find out the answer to "Fonction inverse of f(x)=x+2+x is x4−4x2+44x2"

lugreget9

Answered question

2022-01-15

Fonction inverse of

f (x) = \sqrt{x + 2} + \sqrt{x}

\frac{x^{4} - 4 x^{2} + 4}{4 x^{2}}

Answer & Explanation

sonSnubsreose6v

Beginner2022-01-16Added 21 answers

Step 1
With

x \geq 0

and

y \geq \sqrt{2}

{(y - \sqrt{x})}^{2}

y^{2} - 2 \sqrt{x} y + x

= {(\sqrt{x + 2})}^{2}

= x + 2

gives

\sqrt{x} = \frac{y^{2} - 2}{2 y}

temzej9

Beginner2022-01-17Added 30 answers

Step 1
Per the request of the OP, here is solution based directly on his approach. His last line is:

\frac{x}{2} - 1

= y + \sqrt{(y + 2) y}

= \sqrt{y} (\sqrt{y} + \sqrt{y + 2})

= \sqrt{y} \times x

Squaring both sides:

\frac{{(x^{2} - 2)}^{2}}{4} = x^{2} y

or:

\frac{{(x^{2} - 2)}^{2}}{4 x^{2}} = y

alenahelenash

Expert2022-01-24Added 556 answers

Step 1 Have reached a good point:

\frac{x^{2}}{2} - 1

= \sqrt{(y + 2)} y \Rightarrow \frac{x^{2}}{2} - 1 - y

= \sqrt{(y + 2) y} \Rightarrow \frac{x^{4}}{4} + 1 + y^{2} - x^{2} - x^{2} y + 2 y

= y^{2} + 2 y

Here, we have simply squared both sides, knowing that LHS and RHS are always greater or equal to 0 with your contrainst. Now, we have to solve for y. We obtain:

\Rightarrow \frac{x^{4}}{4} + 1 - x^{2} - x^{2} y

= 0 \Rightarrow \frac{x^{4} + 4 - 4 x^{2}}{4}

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

= \frac{x^{4} - 4 x^{2} + 4}{4 x^{2}}

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