Let's work through both parts (a) and (b) of the question.

(a) Find a formula for the inverse of $f(x) = 1 + \sqrt{4 + 5x}$.

To find the inverse of a function, we follow these steps:

1. Rewrite the function as $y = 1 + \sqrt{4 + 5x}$.

2. Swap $x$ and $y$ because we are solving for the inverse function. $x = 1 + \sqrt{4 + 5y}$

3. Solve for $y$:

   • Subtract 1 from both sides: $x - 1 = \sqrt{4 + 5y}$
   • Square both sides to remove the square root: $(x - 1)^2 = 4 + 5y$
   • Subtract 4 from both sides: $(x - 1)^2 - 4 = 5y$
   • Divide both sides by 5: $y = \frac{(x - 1)^2 - 4}{5}$

Thus, the formula for the inverse is: $f^{-1}(x) = \frac{(x - 1)^2 - 4}{5}$

(b) For what values of $x$ is $f^{-1}(x)$ defined?

To determine where the inverse function is defined, we need to consider the domain of the original function $f(x)$, which becomes the range of the inverse function.

• The square root function $\sqrt{4 + 5x}$ must have a non-negative argument. Thus, $4 + 5x \geq 0$, which simplifies to: $x \geq -\frac{4}{5}$ This gives the domain of $f(x)$, meaning that the range of the inverse function $f^{-1}(x)$ corresponds to values where: $x \geq 1$ since $f(x) = 1 + \sqrt{4 + 5x}$ has a minimum value of 1.

Thus, $f^{-1}(x)$ is defined for $x \geq 1$.

Do you need further details or have any questions?

Here are some relative questions to further explore:

1. How would the process change if the function involved a different type of square root, such as $\sqrt{5x - 4}$?
2. What is the significance of squaring both sides when finding the inverse of a function involving square roots?
3. How can we verify that the formula for the inverse is correct?
4. What are some general strategies for finding the domain and range of inverse functions?
5. Can you find the inverse for a function involving cube roots or higher-degree roots?

Tip:

When finding the inverse of a function, always make sure to check if the inverse function satisfies the original function's domain restrictions.