Question
Consider the function \( f(x)=\sqrt{4-x} \). Find the formula for the inverse function.
Ask by Montgomery Santiago. in the United States
Jan 26,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The inverse function is \( f^{-1}(x) = 4 - x^2 \).
Solution
To find the inverse of the function \( f(x) = \sqrt{4 - x} \), follow these steps:
1. **Start with the original equation:**
\[
y = \sqrt{4 - x}
\]
2. **Solve for \( x \) in terms of \( y \):**
\[
y^2 = 4 - x
\]
\[
x = 4 - y^2
\]
3. **Express the inverse function:**
\[
f^{-1}(y) = 4 - y^2
\]
Typically, we replace \( y \) with \( x \) to express the inverse function in standard form:
\[
f^{-1}(x) = 4 - x^2
\]
**Domain and Range Considerations:**
- **Original Function \( f(x) = \sqrt{4 - x} \):**
- **Domain:** \( x \leq 4 \) (since the expression inside the square root must be non-negative)
- **Range:** \( y \geq 0 \) (since square roots are non-negative)
- **Inverse Function \( f^{-1}(x) = 4 - x^2 \):**
- **Domain:** \( x \geq 0 \) (based on the range of the original function)
- **Range:** \( y \leq 4 \)
**Final Answer:**
\[
f^{-1}(x) = 4 - x^2
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the inverse function of \( f(x) = \sqrt{4 - x} \), we start by setting \( y = f(x) \). Therefore, we have: \[ y = \sqrt{4 - x} \] Next, we swap \( x \) and \( y \): \[ x = \sqrt{4 - y} \] Now, we solve for \( y \). First, we square both sides: \[ x^2 = 4 - y \] Rearranging gives us: \[ y = 4 - x^2 \] Thus, the inverse function is: \[ f^{-1}(x) = 4 - x^2 \] And remember, this inverse is only valid for \( x \) in the range that corresponds to the domain of the original function \( f \). So, the formula for the inverse function is: \[ f^{-1}(x) = 4 - x^2 \]
