Let's analyze and solve the given questions:

Question #4: Graph Analysis

1. Find intercept(s), if any:

   • x-intercept(s): The x-intercept(s) occur where the graph crosses the x-axis (where $y = 0$). From the graph, the x-intercept appears to be around $x = 1$.
   • y-intercept(s): The y-intercept occurs where the graph crosses the y-axis (where $x = 0$). From the graph, this is around $y = 3$.

2. Find vertical asymptote(s), if any:

   • Vertical asymptotes occur where the function is undefined, typically where the denominator of a rational function is zero. From the graph, vertical asymptotes appear at $x = -2$ and $x = 4$.

3. Find any horizontal asymptote, if any:

   • Horizontal asymptotes describe the behavior of the function as $x$ approaches positive or negative infinity. From the graph, it appears that the horizontal asymptote is $y = 0$.

4. Use the graph to write an equation for the function:

   • A possible equation that fits the graph and the asymptotes is a rational function like: $f(x) = \frac{a}{(x+2)(x-4)}$ where $a$ is a constant that can be determined using other points on the graph. Since it crosses the y-axis at $y = 3$ when $x = 0$: $f(0) = \frac{a}{(0+2)(0-4)} = 3 \implies \frac{a}{-8} = 3 \implies a = -24$ So, the equation could be: $f(x) = \frac{-24}{(x+2)(x-4)}$

Question #5: Inverse Function and Domain/Range

1. Find the inverse of the function $f(x) = 9 + \sqrt{4x - 4}$:

   • To find the inverse, let $y = 9 + \sqrt{4x - 4}$.
   • Switch $x$ and $y$: $x = 9 + \sqrt{4y - 4}$
   • Solve for $y$: $x - 9 = \sqrt{4y - 4}$ $(x - 9)^2 = 4y - 4$ $y = \frac{(x - 9)^2 + 4}{4}$ Thus, the inverse function is: $f^{-1}(x) = \frac{(x - 9)^2 + 4}{4}$

2. Find the domain and range of both $f(x)$ and $f^{-1}(x)$:

   • Domain of $f(x)$:
     • $f(x)$ involves a square root, so $4x - 4 \geq 0$.
     • This simplifies to $x \geq 1$.
     • Thus, the domain of $f(x)$ is $[1, \infty)$.
   • Range of $f(x)$:
     • Since $f(x)$ includes $9 + \sqrt{4x - 4}$, the smallest value occurs at $x = 1$: $f(1) = 9 + \sqrt{4(1) - 4} = 9$
     • As $x \to \infty$, $f(x) \to \infty$.
     • Thus, the range of $f(x)$ is $[9, \infty)$.
   • Domain of $f^{-1}(x)$:
     • The domain of the inverse function is the range of $f(x)$, so it is $[9, \infty)$.
   • Range of $f^{-1}(x)$:
     • The range of the inverse function is the domain of $f(x)$, so it is $[1, \infty)$.

Would you like further details or explanations on any part of this solution?

Related Questions:

1. How do you determine the vertical asymptotes of a rational function?
2. What is the significance of a horizontal asymptote in a function's graph?
3. How do you verify if two functions are inverses of each other?
4. Why is it important to consider the domain and range when finding an inverse function?
5. What changes occur to a graph when a function is transformed (e.g., shifts or stretches)?

Tip:

When finding the inverse of a function involving square roots, always remember to consider the domain and range restrictions, as they ensure the inverse is a proper function.