Problem 26 - Algebra 2

Question

For each function \(f,\) find \(f^{-1}\) and the domain and range of \(f\) and \(f^{-1} .\) Determine whether \(f^{-1}\) is a function. $$ f(x)=\sqrt{-2 x+3} $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=\sqrt{-2x+3}\) is \(f^{-1}(y)=\frac{3-x^2}{2}\). The domain and range of the original function are \(x \in (-\infty,\frac{3}{2}]\) and [0,\infty), whereas the domain and range of the inverse function are [0,\infty) and \(y \in (-\infty,\frac{3}{2}]\) respectively. It has been verified that \(f^{-1}\) is a function.

Step 1: Finding the inverse function

To find the inverse of \(f(x)=\sqrt{-2x+3}\), we first swap 'x' and 'y', to get \(x=\sqrt{-2y+3}\). Next, square both sides of the equation to remove the square root, resulting in \(x^2=-2y+3\). Finally, we rearrange the equation for y to find the inverse function, leading to \(f^{-1}(y)=\frac{3-x^2}{2}\).

Step 2: Find the domain and range of f

The domain of \(f(x)=\sqrt{-2x+3}\) are all real values for which the value under the square root is non-negative. This means, \(-2x+3\ge 0\), leading to \(x\le \frac{3}{2}\). Hence the domain of f is \(x\in (-\infty,\frac{3}{2}]\). The range of a square root function is always \([0,\infty)\), as square root always delivers a non-negative output.

Step 3: Find the domain and range of \(f^{-1}\)

The domain of the inverse function is always the range of the initial function, hence for \(f^{-1}(y)=\frac{3-x^2}{2}\), the domain is \(y \in [0,\infty)\). For the range, as \(-x^2\) will always result in a value less than or equal to zero, the maximum value of the numerator \(3-x^2\) is 3 and the range is \(y \in (-\infty,\frac{3}{2}]\)

Step 4: Verifying if \(f^{-1}\) is a function

By inspecting the equation for the inverse \(f^{-1}(y)=\frac{3-x^2}{2}\), we can see it passes the vertical line test, meaning for each x value there is exactly one corresponding y value, confirming that \(f^{-1}\) is indeed a function

Key Concepts

Domain and Range

Understanding the domain and range of a function helps determine the set of possible inputs and outputs. In everyday terms, the domain includes values that you can plug into the function without causing any mathematical issues, such as dividing by zero or taking the square root of a negative number.
For the function \( f(x) = \sqrt{-2x+3} \), the domain requires the expression inside the square root, \(-2x + 3\), to be non-negative. Solving for \( x \) gives \(-2x + 3 \geq 0\), so the domain is all \( x \) that satisfy \( x \leq \frac{3}{2} \).
The range of a function is the set of all possible output values. Since square root functions only produce non-negative values, the range of \( f(x) \) is \([0, \infty)\). Understanding these limits helps predict the behavior of a function, guiding you in solving problems efficiently.

Square Root Function

Square root functions play a crucial role in mathematics, modeling situations where quantities increase slowly at first and more rapidly later on. The standard form is \( y = \sqrt{x} \), but they can appear as part of more complex expressions too.
With \( f(x) = \sqrt{-2x+3} \), the function has been transformed compared to the basic \( \sqrt{x} \). This involves reflecting the function over certain axes or shifting it along them, impacting how the function behaves depending on input \( x \). These transformations affect the domain as previously discussed, but not the core idea that outputs remain non-negative.
Understanding how to handle these variations is pivotal. It ensures you can untangle complex-looking expressions into simpler parts, making it easier to solve or invert functions.

Function Notation

Function notation is crucial for understanding and communicating mathematical concepts succinctly. When you see \( f(x) \), it signifies that \( x \) is the variable input into the function \( f \). Output depends on the specific rule or equation given by \( f \).
In the inverse function \( f^{-1}(x) \), the notation signifies a process to reverse the role of inputs and outputs of \( f \). If \( f(x) \) represents a transformation taking \( x \) to \( y \), then \( f^{-1}(x) \) does the opposite, aiming to recover \( x \) from \( y \).
Employing function notation allows for concise, precise expressions, facilitating problem-solving by providing a clear designation of roles each variable assumes in calculations. This notation is foundational for communicating and solving equations effectively in mathematics.