To find the inverse function \( h^{-1}(x) \), we need to solve the equation \( y = \sqrt{3x + 8} + 13 \) for \( x \) in terms of \( y \). This involves isolating \( x \) on one side of the equation. Once we have the inverse function, we can determine the domain and range of both \( h(x) \) and \( h^{-1}(x) \). The domain of \( h(x) \) is determined by the values of \( x \) for which the expression under the square root is non-negative, and the range is determined by the output values of \( h(x) \). The domain and range of \( h^{-1}(x) \) are the range and domain of \( h(x) \), respectively.
To find the inverse function \( h^{-1}(x) \), we start by setting \( y = h(x) \). Thus, we have:
\[
y = \sqrt{3x + 8} + 13
\]
To find the inverse, we need to solve for \( x \) in terms of \( y \). First, isolate the square root:
\[
y - 13 = \sqrt{3x + 8}
\]
Next, square both sides to eliminate the square root:
\[
(y - 13)^2 = 3x + 8
\]
Now, solve for \( x \):
\[
3x = (y - 13)^2 - 8
\]
\[
x = \frac{(y - 13)^2 - 8}{3}
\]
Thus, the inverse function is:
\[
h^{-1}(x) = \frac{(x - 13)^2 - 8}{3}
\]
The function \( h(x) = \sqrt{3x + 8} + 13 \) is defined when the expression under the square root is non-negative:
\[
3x + 8 \geq 0
\]
Solving for \( x \):
\[
3x \geq -8
\]
\[
x \geq -\frac{8}{3}
\]
Thus, the domain of \( h(x) \) is:
\[
\left[-\frac{8}{3}, \infty\right)
\]
Since \( h(x) = \sqrt{3x + 8} + 13 \), the minimum value of \( \sqrt{3x + 8} \) is 0 (when \( x = -\frac{8}{3} \)). Therefore, the minimum value of \( h(x) \) is:
\[
0 + 13 = 13
\]
As \( x \to \infty \), \( \sqrt{3x + 8} \to \infty \), so \( h(x) \to \infty \).
Thus, the range of \( h(x) \) is:
\[
[13, \infty)
\]
The domain of the inverse function \( h^{-1}(x) \) is the range of the original function \( h(x) \). From Step 3, we have:
\[
[13, \infty)
\]
The range of the inverse function \( h^{-1}(x) \) is the domain of the original function \( h(x) \). From Step 2, we have:
\[
\left[-\frac{8}{3}, \infty\right)
\]
- Inverse function: \(\boxed{h^{-1}(x) = \frac{(x - 13)^2 - 8}{3}}\)
- Domain of \( h(x) \): \(\boxed{\left[-\frac{8}{3}, \infty\right)}\)
- Domain of \( h^{-1}(x) \): \(\boxed{[13, \infty)}\)
- Range of \( h(x) \): \(\boxed{[13, \infty)}\)
- Range of \( h^{-1}(x) \): \(\boxed{\left[-\frac{8}{3}, \infty\right)}\)
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