Question

function: $g(x) = 2x^2 - 8$
for $x \geq 0$, the inverse function is $f(x)= \sqrt{\frac{1}{2}x + 4}$
for $x \leq 0$, the inverse function is $d(x)= - \sqrt{\frac{1}{2}x + 4}$

$x$	$f(x)$	$d(x)$
$0$	$r$	$-2$
$10$	$s$	$t$

$q = \square$
$r = \square$
$s = \square$
$t = \square$

Explanation:

Step1: Find q (x = -8, use d(x))

We use the inverse function \( d(x)=-\sqrt{\frac{1}{2}x + 4} \) with \( x=-8 \).

$$ d(-8)=-\sqrt{\frac{1}{2}(-8)+4}=-\sqrt{-4 + 4}=-\sqrt{0}=0 $$

So \( q = 0 \).

Step2: Find r (x = 0, use f(x))

We use the inverse function \( f(x)=\sqrt{\frac{1}{2}x + 4} \) with \( x = 0 \).

$$ f(0)=\sqrt{\frac{1}{2}(0)+4}=\sqrt{0 + 4}=\sqrt{4}=2 $$

So \( r = 2 \).

Step3: Find s (x = 10, use f(x))

We use the inverse function \( f(x)=\sqrt{\frac{1}{2}x + 4} \) with \( x = 10 \).

$$ f(10)=\sqrt{\frac{1}{2}(10)+4}=\sqrt{5 + 4}=\sqrt{9}=3 $$

So \( s = 3 \).

Step4: Find t (x = 10, use d(x))

We use the inverse function \( d(x)=-\sqrt{\frac{1}{2}x + 4} \) with \( x = 10 \).

$$ d(10)=-\sqrt{\frac{1}{2}(10)+4}=-\sqrt{5 + 4}=-\sqrt{9}=-3 $$

So \( t = -3 \).

Answer:

\( q = 0 \)
\( r = 2 \)
\( s = 3 \)
\( t = -3 \)