Jika f(x) = root((1 - x) ^ 3, 5) + 5 maka f^-1(4) =

To find \( f^{-1}(4) \), we need to determine the value of \( x \) such that \( f(x) = 4 \).

Given the function:

\[
f(x) = \sqrt[5]{(1 - x)^3} + 5
\]

We want to find \( x \) such that:

\[
f(x) = 4
\]

### Step-by-Step Solution:

1. **Set up the equation**:

   \[
   \sqrt[5]{(1 - x)^3} + 5 = 4
   \]

2. **Isolate the radical term**:

   \[
   \sqrt[5]{(1 - x)^3} = 4 - 5
   \]

   \[
   \sqrt[5]{(1 - x)^3} = -1
   \]

3. **Raise both sides to the power of 5 to eliminate the fifth root**:

   \[
   (1 - x)^3 = (-1)^5
   \]

   Since \((-1)^5 = -1\), we get:

   \[
   (1 - x)^3 = -1
   \]

4. **Take the cube root of both sides**:

   \[
   1 - x = \sqrt[3]{-1}
   \]

   Since \(\sqrt[3]{-1} = -1\), we have:

   \[
   1 - x = -1
   \]

5. **Solve for \( x \)**:

   \[
   x = 1 + 1
   \]

   \[
   x = 2
   \]

### Conclusion:

\[
f^{-1}(4) = 2
\]

The inverse function value \( f^{-1}(4) \) is \( 2 \).

Would you like more details or have any questions about this solution?

Here are 5 related questions to consider:
1. What is the domain and range of the original function \( f(x) = \sqrt[5]{(1 - x)^3} + 5 \)?
2. How do you determine if a function has an inverse that is also a function?
3. How do you find the inverse of other functions involving roots and exponents?
4. What is the graphical interpretation of finding \( f^{-1}(4) \)?
5. How would the process change if the function involved a different root or exponent?

**Tip:** When working with inverse functions, ensure the function is one-to-one or restrict its domain to make it invertible.

Discover how to find the inverse function value f^-1(4) for the function f(x) = sqrt[5]{(1 - x)^3} + 5 with a step-by-step solution. This problem involves inverse functions, fifth roots, and algebraic manipulation. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation