Exercise 2 | Wiskunde op Tilburg University

Consider the function

$y(x)=2x^2 + 4x - 2$ ,

$(x\leq-1)$ . Determine the inverse function

$x(y)$ .

Antwoord 1 correct

Correct

Antwoord 2 optie

$x(y) = -1 + \sqrt{8+2y}$ .

Antwoord 2 correct

Fout

Antwoord 3 optie

$x(y) = -1 \pm \sqrt{8+2y}$ .

Antwoord 3 correct

Fout

Antwoord 4 optie

The inverse function of

$y(x)$ cannot be determined.

Antwoord 4 correct

Fout

Antwoord 1 optie

$x(y) = -1 - \sqrt{8+2y}$ .

Antwoord 1 feedback

Correct: We can rewrite

$y = 2x^2 + 4x - 2$ by the use of the Discriminant criterion (

$(*)$ ):

$\begin{align*} y &= 2x^2 + 4x - 2\\ 0 &= 2x^2 + 4x - 2 - y\\ x &\stackrel{(*)}{=} \dfrac{-4 \pm \sqrt{4^2 - 4\cdot 2\cdot (-2-y)}}{2\cdot 2} = \dfrac{-4 \pm \sqrt{16 - 8(-2-y)}}{4} = \dfrac{-4 \pm \sqrt{32 + 8y}}{4} = \dfrac{-4 \pm \sqrt{4}\sqrt{8 + 2y}}{4} \\ &= -1 \pm \tfrac{1}{2}\sqrt{8 + 2y}. \end{align*}$
Since it is given that

$x\leq -1$ , we know that the inverse function is equal to

$x(y) = -1 - \tfrac{1}{2}\sqrt{8+2y}.$

Go on.

Antwoord 2 feedback

Wrong: Consider the domain of

$y(x)$ .

See Example 2.

Antwoord 3 feedback

Wrong: The inverse function has only one output, not two. Consider the domain of

$y(x)$ .

See Example 2.

Antwoord 4 feedback

Wrong: It is possible to determine the inverse function if you use the Discriminant criterion.

See Example 2.