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.
$$\begin{align*}
y &= 2x^2 + 4x - 2\\
0 &= 2x^2 + 4x - 2 - y\\
x &\stackrel{(*)}{=} \dfrac{-4 \pm \sqrt{4^2 - 4\cdot 2\cdot (-2-y)}}{2\cdot 2} = \dfrac{-4 \pm \sqrt{16 - 8(-2-y)}}{4} = \dfrac{-4 \pm \sqrt{32 + 8y}}{4} = \dfrac{-4 \pm \sqrt{4}\sqrt{8 + 2y}}{4} \\
&= -1 \pm \tfrac{1}{2}\sqrt{8 + 2y}.
\end{align*}
$$
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
Antwoord 3 feedback
Wrong: The inverse function has only one output, not two. Consider the domain of $y(x)$.
See Example 2.
See Example 2.
Antwoord 4 feedback
Wrong: It is possible to determine the inverse function if you use the Discriminant criterion.
See Example 2.
See Example 2.