To identify the domain and range of the inverse function $f^{-1}(x)$, we first need to understand the domain and range of the original function $f(x)$.

Given the function $f(x) = 7x^2 + 42x + 71$ with the domain $-3 \leq x \leq 5$:

1. Domain of $f(x)$: The domain is the set of all input values $x$ for the function. Here, it is given directly as $-3 \leq x \leq 5$.

2. Range of $f(x)$: The range is the set of all possible output values $y = f(x)$ within the given domain. To find the range, we need to evaluate $f(x)$ at the endpoints and at the vertex of the parabola, as the function is quadratic (a parabola).

   • Finding the vertex: The vertex $x$-coordinate for a parabola $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$. Here, $a = 7$ and $b = 42$. $x = -\frac{42}{2(7)} = -3$ Since the vertex occurs at $x = -3$, this value lies within the domain.

   • Calculating the function value at $x = -3$ (vertex): $f(-3) = 7(-3)^2 + 42(-3) + 71 = 7(9) - 126 + 71 = 63 - 126 + 71 = 8$

   • Calculating the function value at $x = 5$: $f(5) = 7(5)^2 + 42(5) + 71 = 7(25) + 210 + 71 = 175 + 210 + 71 = 456$

   So, the range of $f(x)$ is $[8, 456]$.

3. Domain and Range of $f^{-1}(x)$:

   • The domain of $f^{-1}(x)$ is the range of $f(x)$, so $\text{Domain of } f^{-1}(x) = [8, 456]$.
   • The range of $f^{-1}(x)$ is the domain of $f(x)$, so $\text{Range of } f^{-1}(x) = [-3, 5]$.

Summary:

• Domain of $f^{-1}(x)$: $[8, 456]$
• Range of $f^{-1}(x)$: $[-3, 5]$

Would you like a detailed explanation or have any other questions?

Here are five questions related to this problem:

1. How do you derive the formula for the vertex of a quadratic function?
2. What is the significance of finding the vertex in the context of this problem?
3. How would the domain and range change if the interval for $x$ was different?
4. What would happen to the domain and range of $f^{-1}(x)$ if the parabola opened downwards?
5. Can the inverse of any quadratic function always be found?

Tip: Always double-check the vertex calculations when dealing with quadratic functions, as they determine the range for the function and hence the domain of its inverse.