Problem 72 - College Algebra with Modeling and Visualization

Question

Find a formula for \(f^{-1}(x) .\) Identify the domain and range of \(f^{-1}\). Verify that \(f\) and \(f^{-1}\) are inverses. $$ f(x)=(x+3)^{2}, x \geq-3 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt{x} - 3 \). The domain is \( x \geq 0 \) and the range is \( x \geq -3 \).

Step 1: Recognize the Conditions

The function given is \( f(x) = (x+3)^2 \) with the domain \( x \geq -3 \). This is a quadratic function with a horizontal shift to the left by 3 units.

Step 2: Express the Function in Terms of Another Variable

Since \( f(x) = y \), we rewrite the function as \( y = (x+3)^2 \). Our goal is to solve this equation for \( x \) to find the inverse.

Step 3: Solve for \( x \) in Terms of \( y \)

Start by taking the square root of both sides: \( \sqrt{y} = x + 3 \). Then isolate \( x \) by subtracting 3 from both sides to get \( x = \sqrt{y} - 3 \).

Step 4: Write the Inverse's Formula

Thus, the inverse function is \( f^{-1}(x) = \sqrt{x} - 3 \).

Step 5: Determine the Domain of \( f^{-1} \)

Since \(f(x)\) maps \(x \geq -3\) to \( y \geq 0 \), the range of \( f(x) \) becomes the domain of \( f^{-1}(x) \). Hence, the domain of \( f^{-1} \) is \( x \geq 0 \).

Step 6: Determine the Range of \( f^{-1} \)

The range of the inverse function corresponds to the domain of the original function, so the range of \( f^{-1}(x) \) is \( x \geq -3 \).

Step 7: Verify that \( f \) and \( f^{-1} \) are Inverses

To verify, we need to check if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \):1. Compute \( f(f^{-1}(x)) = f(\sqrt{x} - 3) = ((\sqrt{x} - 3) + 3)^2 = x \).2. Compute \( f^{-1}(f(x)) = f^{-1}((x+3)^2) = \sqrt{(x+3)^2} - 3 = x \) since \( x \geq -3 \).Both computations return \( x \), thereby verifying they are inverses.

Key Concepts

Domain and Range

When we talk about the domain and range of functions, we are referring to the possible inputs (domain) and potential outputs (range) that a function can have. For the function given in the exercise, we start with \( f(x) = (x+3)^2 \) where \( x \geq -3 \). This function's domain, therefore, allows any number greater than or equal to \(-3\) to be used as input.

Because the function is a quadratic one, it transforms its domain into non-negative outputs. Hence, the range of \( f(x) \) is \( y \geq 0 \).

• The domain refers to all possible input values, for this function, it is \( x \geq -3 \).
• The range corresponds to all output values, which in this case is \( y \geq 0 \).

To find the inverse function, \( f^{-1}(x) \), we switch the roles. The range of \( f(x) \), which was \( y \geq 0 \), becomes the domain of \( f^{-1}(x) \). Therefore, \( f^{-1}(x) \)'s domain is \( x \geq 0 \). Similarly, the original domain \( x \geq -3 \) becomes the range of \( f^{-1}(x) \).

Quadratic Functions

Quadratic functions are a category of polynomial functions where the highest degree of any term is two. In simpler terms, they are functions where the variable is squared, leading to a characteristic "U" shape graph called a parabola.

The function in question, \( f(x) = (x+3)^2 \), is a quadratic function. Its graph is a parabola opening upwards, and it has been shifted horizontally to the left by 3 units compared to the standard \( y = x^2 \) function. This shift is due to the \( (x+3) \) in the function, indicating the parabola’s vertex is at \((-3, 0)\).

• The graph of a quadratic function is a parabola.
• For \( f(x) = (x+3)^2 \), the vertex of the parabola is shifted left by 3 units.

Understanding the behavior of parabolas is helpful because they show us potential ranges of values and key turning points, like vertices, that play a role in defining inverse functions.

Function Verification

Verifying if two functions are inverses involves proving they can "undo" each other. This means if you apply both functions in succession, you should arrive back at the starting value. It involves two steps:

1. Checking \( f(f^{-1}(x)) = x \)
2. Ensuring \( f^{-1}(f(x)) = x \)

Let's verify:\( f(f^{-1}(x)) = f(\sqrt{x} - 3) \). Substituting, we get \( ((\sqrt{x} - 3) + 3)^2 = x \).

For \( f^{-1}(f(x)) = f^{-1}((x+3)^2) \), substituting gives us \( \sqrt{(x+3)^2} - 3 = x \). Note that because \( x \geq -3 \), the square root operation honors positive solutions, confirming \( x \).

• To verify inverse functions, both \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) should equal \( x \).
• For this exercise, doing both operations confirms they return \( x \).

This ensures they are truly inverse functions, completing our function verification.