Problem 77 - Algebra and Trigonometry

Question

The given function is not one-to-one. Restrict its domain so that the resulting function \(i s\) one-to-one. Find the inverse of the function with the restricted domain. (There is more than one correct answer.) $$ h(x)=(x+2)^{2} $$

Step-by-Step Solution

Answer

Restrict domain to \( x \geq -2 \) and inverse is \( h^{-1}(x) = \sqrt{x} - 2 \).

Step 1: Identify the Problem with One-to-One

The function \( h(x) = (x+2)^2 \) is not one-to-one because it is a quadratic function with the horizontal line test indicating that there are duplicate y-values for different x-values. A function must pass the horizontal line test to be one-to-one such that each output is only associated with one input.

Step 2: Restrict the Domain

To make \( h(x) = (x+2)^2 \) one-to-one, we restrict the domain. Since the squared function is symmetric, we can choose either the left side (\( x \leq -2 \)) or the right side (\( x \geq -2 \)). For this solution, we will use the domain \( x \geq -2 \).

Step 3: Find the Inverse Function

To find the inverse, start with the equation \( y = (x+2)^2 \) and solve for \( x \). First, take the square root of both sides (considering only the positive root due to the restricted domain \( x \geq -2 \)), hence \( \sqrt{y} = x + 2 \). Isolate \( x \) to get \( x = \sqrt{y} - 2 \). Therefore, the inverse function is \( h^{-1}(x) = \sqrt{x} - 2 \) with the domain \( x \geq 0 \).

Step 4: Verify the Inverse

Verify the inverse by checking if \( h(h^{-1}(x)) = x \) and \( h^{-1}(h(x)) = x \). Check: 1) \( h(h^{-1}(x)) = ((\sqrt{x} - 2) + 2)^2 = x \), valid for \( x \geq 0 \). 2) \( h^{-1}(h(x)) = \sqrt{(x+2)^2} - 2 = x \), valid for \( x \geq -2 \). Hence, both function compositions return \( x \), verifying correctness.

Key Concepts

One-to-One Function

In mathematics, a function is considered "one-to-one" if each element of the range is paired with exactly one element of the domain. This means no two different input values produce the same output. One way to determine if a function is one-to-one is through the "horizontal line test". If any horizontal line crosses the graph of a function more than once, the function fails the test and is not one-to-one. Quadratic functions, like the one given in our exercise \( h(x) = (x+2)^2 \), tend to fail this test because they are symmetrical. As a result, multiple \( x \)-values can produce the same \( y \)-value. For example, with \( h(x) = (x+2)^2 \), both \( x = -1 \) and \( x = -3 \) would give \( h(x) = 1 \). That's why it's crucial to restrict the domain of a quadratic to make it one-to-one.

Domain Restriction

A domain restriction helps in assessing a function's one-to-one property. By restricting the domain, we can ensure that each output value is the result of one unique input value, thereby fulfilling the one-to-one condition. For the function \( h(x) = (x+2)^2 \), we chose to restrict the domain to \( x \geq -2 \). This choice limits the \( x \)-values to only those that are greater than or equal to -2, effectively using only the right side of the parabola.

• The domain \( x \geq -2 \) ensures every \( y \)-value is unique because each value of \( y \) is generated from just one \( x \).
• The domain can also be restricted on the left side \( x \leq -2 \), achieving the same one-to-one requirement, showing flexibility in problem-solving.

Understanding and choosing the right domain is key in finding the inverse of a function, as it directly affects the range of the inverse.

Inverse Function Verification

Once a function becomes one-to-one through domain restriction, its inverse can be computed with confidence. It's paramount, however, to verify that the inverse function is correct. For the function \( h(x) = (x+2)^2 \), after restricting the domain, we derived the inverse function as \( h^{-1}(x) = \sqrt{x} - 2 \) with domain \( x \geq 0 \).

• The verification process involves checking if both \( h(h^{-1}(x)) = x \) and \( h^{-1}(h(x)) = x \) hold true.
• First, substitute \( x \) into \( h^{-1}(x) \) and plug this into \( h(x) \), which results in the original \( x \), proving the inverse is correct for \( x \geq 0 \).
• Similarly, substituting \( x \) into \( h(x) \) and then \( h^{-1}(x) \) confirms that the compositions yield original \( x \), thus verifying the inverse function correctly models the original function's behavior.

Verifying the inverse ensures that we've accurately translated the function's transformations, safeguarding against logical errors.