- We use the given inverse function formulas \(f(x) = \sqrt{\frac{1}{2}x + 4}\) and \(d(x) = -\sqrt{\frac{1}{2}x + 4}\).
- Calculate \(q = d(-8) = -\sqrt{\frac{1}{2}(-8) + 4} = 0\).
- Calculate \(r = f(0) = \sqrt{\frac{1}{2}(0) + 4} = 2\).
- Calculate \(s = f(10) = \sqrt{\frac{1}{2}(10) + 4} = 3\).
- Calculate \(t = d(10) = -\sqrt{\frac{1}{2}(10) + 4} = -3\).
- Final values are \(q = \boxed{0}, r = \boxed{2}, s = \boxed{3}, t = \boxed{-3}\).
Explanation
- Problem Analysis and Definitions
We are given the function \(g(x) = 2x^2 - 8\). We identify the inverse functions defined by splitting the domain of \(g(x)\):
- For \(x \geq 0\), the inverse is \(f(x) = \sqrt{\frac{1}{2}x + 4}\).
- For \(x \leq 0\), the inverse is \(d(x) = -\sqrt{\frac{1}{2}x + 4}\).
We need to determine the missing values \(q, r, s,\) and \(t\) by evaluating these functions at specific input values given in the table.
- Calculating q
The value \(q\) corresponds to the function \(d(x)\) evaluated at \(x = -8\).
Using the formula for \(d(x)\):
\[d(-8) = -\sqrt{\frac{1}{2}(-8) + 4}\]
\[d(-8) = -\sqrt{-4 + 4}\]
\[d(-8) = -\sqrt{0}\]
\[q = 0\]
- Calculating r
The value \(r\) corresponds to the function \(f(x)\) evaluated at \(x = 0\).
Using the formula for \(f(x)\):
\[f(0) = \sqrt{\frac{1}{2}(0) + 4}\]
\[f(0) = \sqrt{0 + 4}\]
\[f(0) = \sqrt{4}\]
\[r = 2\]
- Calculating s
The value \(s\) corresponds to the function \(f(x)\) evaluated at \(x = 10\).
Using the formula for \(f(x)\):
\[f(10) = \sqrt{\frac{1}{2}(10) + 4}\]
\[f(10) = \sqrt{5 + 4}\]
\[f(10) = \sqrt{9}\]
\[s = 3\]
- Calculating t
The value \(t\) corresponds to the function \(d(x)\) evaluated at \(x = 10\).
Using the formula for \(d(x)\):
\[d(10) = -\sqrt{\frac{1}{2}(10) + 4}\]
\[d(10) = -\sqrt{5 + 4}\]
\[d(10) = -\sqrt{9}\]
\[t = -3\]
- Conclusion
Based on the calculations performed above, the values are:
- \(q = 0\)
- \(r = 2\)
- \(s = 3\)
- \(t = -3\)
Thus, the final results are \(q = \boxed{0}\), \(r = \boxed{2}\), \(s = \boxed{3}\), and \(t = \boxed{-3}\).
Examples
Inverse functions are essential in fields like physics and engineering. For example, if you have a formula that relates the distance an object has fallen to the time elapsed, such as \(d(t) = \frac{1}{2}gt^2\), you can use the inverse function to calculate the time taken to fall a certain distance: \(t(d) = \sqrt{\frac{2d}{g}}\). Understanding how to handle domain restrictions when finding inverses is crucial for these calculations, just as we did here by splitting the function into two parts based on the domain.