Problem 6 - College Algebra

Question

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x+2)^{2},[-2, \infty) $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x} - 2 \), with domain \( x \geq 0 \).

Step 1: Understand the function and its domain

The exercise gives us the function \( f(x) = (x+2)^2 \), with a domain \([-2, \infty)\). This means that \( x \) can take any value from \(-2\) to positive infinity.

Step 2: Set up the function equation

To find the inverse, we start by considering the function equation as \( y = (x+2)^2 \). To find \( x \) in terms of \( y \), this equation needs to be manipulated appropriately.

Step 3: Isolate the expression containing x

To isolate \( x \), we take the square root of both sides of the equation \( y = (x+2)^2 \). This gives us two possible equations: \( x + 2 = \sqrt{y} \) and \( x + 2 = -\sqrt{y} \). However, due to the domain \([-2, \infty)\), we only consider the positive root \( x + 2 = \sqrt{y} \).

Step 4: Solve for x

Rearrange the equation \( x + 2 = \sqrt{y} \) to solve for \( x \): \( x = \sqrt{y} - 2 \). This is the expression for \( x \) in terms of \( y \).

Step 5: Write the inverse function

Now, substitute \( y \) back to \( x \), switching the roles of \( x \) and \( y \), to get the inverse function: \( f^{-1}(x) = \sqrt{x} - 2 \).

Step 6: Determine the domain of the inverse

The domain of the original function \( f(x) \) gives us the range of the inverse, so \( f^{-1}(x) = \sqrt{x} - 2 \) is defined for \( x \geq 0 \).

Key Concepts

Domain and Range

Understanding domain and range is crucial when working with functions and their inverses. The domain is the set of all possible input values (x-values) for a function. In our exercise, the domain of the function \( f(x) = (x+2)^2 \) is \([-2, \infty)\). This tells us that \( x \) can be any number starting from -2 and going upwards.

The range, on the other hand, is the set of possible output values (y-values) that the function can produce. For our function, since it involves squaring an expression, the smallest output \( y \) can be is \( 0 \) when \( x = -2 \), because squaring any real number results in a non-negative value.

For inverse functions, the domain and range switch roles. The range of the original function becomes the domain of the inverse. So, when finding \( f^{-1}(x) = \sqrt{x} - 2 \), its domain starts from \( 0 \) upwards, reflecting the range of the original function.

Function Manipulation

Function manipulation often involves isolating a variable to find an inverse, which can be thought of as reversing the operations of the original function. In our example, we started with the equation \( y = (x+2)^2 \) and needed to solve for \( x \) in terms of \( y \). This process is about "undoing" the operations applied to \( x \).

First, we dealt with the exponent by taking the square root of both sides. This left us with \( x+2 = \sqrt{y} \) or \( x + 2 = -\sqrt{y} \). However, due to the domain \([-2, \infty)\), we only took the positive root. This is an important step because without considering the domain, we might end up with a function that doesn't map properly back to the original function.

Finally, isolating \( x \) from \( x + 2 = \sqrt{y} \), we solve: \( x = \sqrt{y} - 2 \). This manipulation ensures that we have successfully expressed \( x \) in terms of \( y \), achieving the function's inverse.

Square Root Functions

Square root functions, like the one we derived for the inverse function, have unique properties due to their domain restrictions. In this context, the function \( f^{-1}(x) = \sqrt{x} - 2 \) involves a square root operation, which implies certain conditions must be met by \( x \).

Since the square root function \( \sqrt{x} \) is only defined for non-negative values, this means that \( x \) must be greater than or equal to 0, which sets the domain of \( f^{-1}(x) \). This domain \( [0, \infty) \) can be seen as a result of the mathematical property that negative numbers do not have real square roots in the set of real numbers.

Moreover, square root functions typically map each \( x \) value to one specific \( y \) value, making them one-to-one when considering only the principal (positive) root. This characteristic is what allows us to define an inverse function such as \( f^{-1}(x) = \sqrt{x} - 2 \) with confidence in its operation.