Math Snap

STEP 1

1. We are given the function $f(x) = 7 + \sqrt{x-9}$.
2. We need to find the inverse function $f^{-1}(x)$.
3. We need to determine the domain and range of both $f$ and $f^{-1}$.

STEP 2

1. Determine the domain of $f(x)$.
2. Find the inverse function $f^{-1}(x)$.
3. Determine the domain of $f^{-1}(x)$.
4. Determine the range of $f(x)$.
5. Determine the range of $f^{-1}(x)$.

STEP 3

To find the domain of $f(x) = 7 + \sqrt{x-9}$, we need the expression under the square root to be non-negative:
$$x - 9 \geq 0$$ Solve for $x$:
$$x \geq 9$$ Thus, the domain of $f$ is:
$$[9, \infty)$$

STEP 4

To find the inverse function $f^{-1}(x)$, start by setting $y = f(x)$:
$$y = 7 + \sqrt{x-9}$$ Solve for $x$ in terms of $y$:
$$y - 7 = \sqrt{x-9}$$ Square both sides:
$$(y - 7)^2 = x - 9$$ Solve for $x$:
$$x = (y - 7)^2 + 9$$ Thus, the inverse function is:
$$f^{-1}(x) = (x - 7)^2 + 9$$

STEP 5

The domain of $f^{-1}(x)$ is the range of $f(x)$. From the expression for $f(x) = 7 + \sqrt{x-9}$, the smallest value occurs when $x = 9$, giving:
$$f(9) = 7 + \sqrt{9-9} = 7$$ Since the square root function can take any non-negative value, the range of $f(x)$ is:
$$[7, \infty)$$ Thus, the domain of $f^{-1}$ is:
$$[7, \infty)$$

STEP 6

The range of $f(x)$ was determined in the previous step as:
$$[7, \infty)$$

The range of $f^{-1}(x)$ is the domain of $f(x)$, which is:
$$[9, \infty)$$ The solutions are:
(a) $f^{-1}(x) = (x - 7)^2 + 9$
(b) The domain of $f$ is $[9, \infty)$
(c) The domain of $f^{-1}$ is $[7, \infty)$
(d) The range of $f$ is $[7, \infty)$
(e) The range of $f^{-1}$ is $[9, \infty)$