Question

Find all the inverses associated with the following functions and state their domains. $$f(x)=(x-4)^{2}$$

Short answer

Answer: The inverse functions are \(f^{-1}_1(x) = \sqrt{x} + 4, x\ge 0\) and \(f^{-1}_2(x) = -\sqrt{x} + 4, x\ge 0\). The domain of both functions is \(x\ge 0\).

Step-by-step solution

Replace f(x) with y

Let's replace \(f(x)\) with \(y\): $$y = (x-4)^2$$

Swap x and y coordinates

Now, swap the roles of x and y in the equation: $$x = (y-4)^2$$

Solve for y

Next, solve for y by taking the square root of both sides of the equation: $$\sqrt{x} = \sqrt{(y-4)^2}$$ $$\pm\sqrt{x} = y-4$$ Then, add 4 to both sides: $$y = \pm\sqrt{x} + 4$$

Determine the inverse function

As there are two possible values of \(y\) after solving for \(y\), we have two possible inverse functions: $$f^{-1}_1(x) = \sqrt{x} + 4$$ $$f^{-1}_2(x) = -\sqrt{x} + 4$$

Find the domain of the inverse functions

For both inverse functions, the input \(x\) must be non-negative because square roots are defined only for non-negative numbers: $$f^{-1}_1(x): x\ge 0$$ $$f^{-1}_2(x): x\ge 0$$ The inverse functions are: $$f^{-1}_1(x) = \sqrt{x} + 4, x\ge 0$$ $$f^{-1}_2(x) = -\sqrt{x} + 4, x\ge 0$$

Key concepts

Inverse Functions

Understanding inverse functions is crucial when studying mathematics, particularly in algebra and calculus. An inverse function essentially reverses the action of the original function. If you have a function that takes an input and gives you an output, the inverse function takes the output and returns the original input.

In more formal terms, if you have a function denoted as \(f(x)\), and you find its inverse, represented as \(f^{-1}(x)\), then applying the inverse function to \(f(x)\) will give you back your original input \(x\).

• If \(y = f(x)\), then \(x = f^{-1}(y)\).
• If \(f(f^{-1}(x)) = x\), then \(f^{-1}(f(x)) = x\).

Finding inverse functions typically involves three main steps:

• Replace \(f(x)\) with \(y\).
• Swap \(x\) and \(y\) to create an equation where \(y\) is the subject.
• Solve the new equation for \(y\), which will give you the inverse function.

To ensure the inverse function exists and is a true function, the original function must be one-to-one, meaning it has exactly one output for each input and passes the horizontal line test.

Domain of a Function

The domain of a function is an essential concept that defines the set of all possible input values for which the function is defined. Understanding the domain allows you to appreciate the limitations and the extent of function applicability.

For instance, consider a square root function, where the domain must be restricted to non-negative numbers since the square root of a negative number is not defined within the set of real numbers.

To determine the domain:

• Look for values that make the function undefined, such as division by zero or negative square roots if dealing with real numbers.
• Consider the context or real-world application which may impose further restrictions on the domain.

In the given exercise, since the inverse functions involve the square root, we note that the input \(x\) cannot be negative. Therefore, the domain of both inverse functions is \(x \textgreater= 0\), where \(x\) is any non-negative real number.

Square Root Properties

Square root properties are important when working with equations that involve square roots. Here are a few properties that are particularly useful:

• The square root of a number \(x\), denoted as \(\sqrt{x}\), is a value that, when multiplied by itself, gives \(x\).
• For any non-negative number \(x\), there are two square roots: positive \(\sqrt{x}\) and negative \(-\sqrt{x}\).
• Square roots are defined only for non-negative numbers in the set of real numbers, setting important domain restrictions.

Applying these properties in the exercise provided, we have to account for both the positive and negative square roots, which leads to two potential inverse functions. Furthermore, because the square root property necessitates non-negative inputs, this reflects directly in the domain of the inverse functions, which must be non-negative as well.