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Magazine Chapter 4: Problem 35
Find the inverse function of \(f\). $$f(x)=\sqrt{3-x}$$
Short Answer
Expert verified
The inverse function is \(f^{-1}(x) = 3-x^2\) with domain \([0, \sqrt{3}]\).
Step by step solution
01
Understand the Problem
To find the inverse of a function, we essentially switch the roles of the output and the input. For the function \(f(x) = \sqrt{3-x}\), we need to express \(x\) in terms of \(y\), where \(y = f(x)\).
02
Write the Function as an Equation
Start with \(y = \sqrt{3-x}\). This is the equation representing our function in terms of \(y\) and \(x\).
03
Swap Variables
Interchange \(x\) and \(y\) to begin finding the inverse. The equation becomes \(x = \sqrt{3-y}\).
04
Solve for the New Output
Solve the equation from Step 3 for \(y\). Square both sides to eliminate the square root: \(x^2 = 3-y\).
05
Isolate the New Input
Rearrange the equation from Step 4 to solve for \(y\): \(y = 3-x^2\). This gives us the inverse of the function.
06
Domain and Range Considerations
For \(f(x) = \sqrt{3-x}\), \(x\) must be less than or equal to 3. This implies restrictions on the domain for the inverse function as well. Since \(f(x)\) takes non-negative values (as it is a square root function), the range of its inverse must match that: \([0, \infty)\). This means for \(f^{-1}(x) = 3-x^2\), \(0 \leq x \leq \sqrt{3}\).
07
Write the Inverse Function
Conclude with the inverse function based on the transformations: The inverse of \(f(x) = \sqrt{3-x}\) is \(f^{-1}(y) = 3-y^2\), where \(0 \leq y \leq \sqrt{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Root Function
Square root functions like the one given: \(f(x) = \sqrt{3-x}\) come from taking the square root of expressions. The square root function is common due to its "half-curve" shape, extending only along positive values.
Generally, for a square root function, as in this case, we only consider values where the inside of the square root is non-negative. Why? Because the square root of a negative number isn't defined in the set of real numbers.
Generally, for a square root function, as in this case, we only consider values where the inside of the square root is non-negative. Why? Because the square root of a negative number isn't defined in the set of real numbers.
- The square root function is key to many mathematical applications and is often used in geometry, physics, and calculus.
- It inherently limits inputs to values making the expression inside the square root non-negative, defining the function's domain.
Function Domain and Range
When working with functions, the domain and range help us understand which inputs and outputs are possible.
Let's take a look at the original function \(f(x) = \sqrt{3-x}\):
Let's take a look at the original function \(f(x) = \sqrt{3-x}\):
- **Domain**: To ensure the square root is valid (non-negative), \(3-x\geq0\). Solving this, we get \(x \leq 3\).
- **Range**: Since the square root itself will always yield zero or a positive number, the range is \([0, \, \infty)\).
- **Domain**: \(0 \leq x \leq \sqrt{3}\) since \(x\) is derived from the range of the original function.
- **Range**: \(3-y^2\), as \(y\) now represents the output from the domain of the function.
Swap Variables
Swapping variables is an essential step in finding the inverse of a function.
This involves replacing the original output of the function with the input variable and vice versa. Consider the original function: \(y = \sqrt{3-x}\). Swapping variables gives us \(x = \sqrt{3-y}\).
The logic here is simple:
This involves replacing the original output of the function with the input variable and vice versa. Consider the original function: \(y = \sqrt{3-x}\). Swapping variables gives us \(x = \sqrt{3-y}\).
The logic here is simple:
- By swapping variables, we effectively redefine roles, making the original output the new input.
- This step helps to transition into solving for the inverse function where the output is expressed in terms of a new variable.
Solving Equations
Solving the equation to find an inverse function involves a series of logical moves to isolate and express one variable in terms of another. For function inverses, solving the swapped equation prepares the inverse function.
Take \(x = \sqrt{3-y}\), and see how we solve:
Take \(x = \sqrt{3-y}\), and see how we solve:
- Square both sides to remove the square root, giving \(x^2 = 3 - y\).
- Isolate \(y\) by rearranging: \(y = 3 - x^2\).
- Breaking down complex terms.
- Understanding how to rearrange elements logically.
- Applying algebraic knowledge until you reach a solution.
