Let's solve for the values of [tex]\( q \)[/tex], [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex] step-by-step, considering the given inverse functions [tex]\( f(x) \)[/tex] and [tex]\( d(x) \)[/tex].
### Step-by-Step Solution:
1. For [tex]\( x = -8 \)[/tex]:
- The corresponding value in the table for [tex]\( f(s) \)[/tex] is given as [tex]\( 0 \)[/tex].
- Since [tex]\( x \leq 0 \)[/tex], we use the inverse function [tex]\( d(x) \)[/tex].
- Therefore, [tex]\( f(s) = 0 \)[/tex] corresponds to [tex]\( d(-8) = 0 \)[/tex].
- Since this value already matches our table, we can directly take [tex]\( q = 0 \)[/tex].
2. For [tex]\( x = 0 \)[/tex]:
- For [tex]\( x \geq 0 \)[/tex], we use the inverse function [tex]\( f(x) \)[/tex].
- Therefore, [tex]\( r \)[/tex] is the value of [tex]\( f(0) \)[/tex].
- Let's substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex]:
[tex]\[
f(0) = \sqrt{\frac{1}{2} \cdot 0 + 4} = \sqrt{4} = 2
\][/tex]
- Hence, [tex]\( r = 2 \)[/tex].
3. For [tex]\( x = 10 \)[/tex]:
- For [tex]\( x \geq 0 \)[/tex], we use the inverse function [tex]\( f(x) \)[/tex].
- Therefore, [tex]\( s \)[/tex] is the value of [tex]\( f(10) \)[/tex].
- Let's substitute [tex]\( x = 10 \)[/tex] into the function [tex]\( f(x) = \sqrt{\frac{1}{2} x + 4} \)[/tex]:
[tex]\[
f(10) = \sqrt{\frac{1}{2} \cdot 10 + 4} = \sqrt{5 + 4} = \sqrt{9} = 3
\][/tex]
- Hence, [tex]\( s = 3 \)[/tex].
4. For [tex]\( x = 0 \)[/tex]:
- For [tex]\( x \leq 0 \)[/tex], we use the inverse function [tex]\( d(x) \)[/tex].
- The table value for [tex]\( k \)[/tex] when [tex]\( x = 0 \)[/tex] is given as [tex]\( -2 \)[/tex].
- This directly corresponds to the function [tex]\( d(0) = -2 \)[/tex].
- [tex]\( t \)[/tex] is the provided value in the table, which matches [tex]\( d(0) \)[/tex].
- Hence, [tex]\( t = -2 \)[/tex].
Now, summarizing the solutions:
[tex]\[
q = 0, \quad r = 2, \quad s = 3, \quad t = -2
\][/tex]
