Given f(x)=7x2+42x+71f(x)=7x2+42x+71,     −3≤x≤5-3≤x≤5.

Identify the Domain and Range of f−1(x)f-1(x)


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.

Explore the domain and range of the inverse function f^{-1}(x) derived from the quadratic function f(x) = 7x^2 + 42x + 71 over the interval -3 ≤ x ≤ 5. Understand how to determine these values, including calculations for vertex and function values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation