Problem 67 - 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 the domain to \( x \geq -2 \), the inverse is \( h^{-1}(x) = \sqrt{x} - 2 \).

Step 1: Analyze the Function

The given function is \( h(x) = (x+2)^2 \). This is a quadratic function with a parabola opening upwards, meaning it is not one-to-one because it will fail the horizontal line test.

Step 2: Restrict the Domain

To make \( h(x) = (x+2)^2 \) one-to-one, we can restrict the domain. We can choose a domain such as \( x \geq -2 \) or \( x \leq -2 \), making it either the right or the left part of the parabola, ensuring one-to-one property.

Step 3: Choose a Domain Restriction

Let's choose \( x \geq -2 \) as our domain restriction. On this domain, \( h(x) = (x+2)^2 \) becomes a one-to-one function as it is strictly increasing.

Step 4: Find the Inverse Function

With the restriction \( x \geq -2 \), solve the equation \( y = (x+2)^2 \) for \( x \). Begin by taking the square root of both sides: \( \sqrt{y} = x + 2 \). Then solve for \( x \) to get \( x = \sqrt{y} - 2 \). Thus, the inverse function is \( h^{-1}(y) = \sqrt{y} - 2 \).

Step 5: Confirm the Inverse Function

Verify that \( h(h^{-1}(y)) = y \) and \( h^{-1}(h(x)) = x \) hold true for the restricted domain. Substitute \( y = (x+2)^2 \) and \( x = \sqrt{y} - 2 \) respectively, which show these statements are true, confirming the inverse. Therefore, the inverse is correctly identified as \( h^{-1}(x) = \sqrt{x} - 2 \).

Key Concepts

One-to-One Function

A one-to-one function is special because every output value is paired with exactly one input. Imagine a club where every member's card matches only one unique name. In terms of functions, this means if you draw a horizontal line across its graph, it should touch the graph at just one point everywhere. This distinct pairing ensures each y-value comes from a distinct x-value.
In simpler terms, for a function \(f(x)\) to be one-to-one:

• Each input (x-value) produces a unique output (y-value).
• No y-values are repeated for different x-values.

Understanding whether a function is one-to-one is crucial for determining the possibility of finding an inverse. If the function isn't naturally one-to-one, we often need to apply domain restrictions.

Domain Restriction

Domain restriction is like setting boundaries. In the mathematical world, this involves limiting the x-values that a function can use. Why do we do this? Well, not all functions can possess an inverse unless tweaked with domain restrictions. For instance, consider a quadratic function like \(h(x) = (x + 2)^2\). It isn't one-to-one over its entire set of x-values, because the parabola pattern repeats y-values.
By restricting domains, like choosing only \(x \geq -2\), the function becomes one-to-one, allowing us to find its inverse. Here are key points:

• Domain restrictions help in making a function one-to-one when it's not naturally so.
• They allow for the calculation of an inverse function, which is crucial in reversing outputs back to original inputs.

In the given exercise, choosing the right restriction enables \(h(x)\) to meet the one-to-one requirement.

Quadratic Function

Quadratic functions are equations of the form \(f(x) = ax^2 + bx + c\). They graph as parabolas. The shape is either a frown or a smile, depending on the sign of the coefficient \(a\). With \(h(x) = (x+2)^2\), the graph is a U-shaped parabola opening upwards.
These functions naturally are not one-to-one over their entire domain because their U shapes mean y-values double up. However, they have useful properties:

• Parabolas have a vertex, which is the highest or lowest point based on their orientation.
• They are symmetric about a vertical line that passes through the vertex.

For solving inverse problems, breaking the parabola into sections and focusing on just one half helps manage the one-to-one condition.

Horizontal Line Test

The horizontal line test is a simple method to check if a function is one-to-one just by looking at its graph. If any horizontal line touches the graph more than once, the function is not one-to-one.
Here's how the test works:

• Draw horizontal lines across the graph.
• If any line intersects the graph at more than one point, the function fails the test.

In the case of \(h(x) = (x+2)^2\), which forms a parabola, horizontal lines drawn will cut through it twice, showing it's not one-to-one. This is why understanding and using the horizontal line test is important in the inverse function exercise: it helps spot the need for domain restriction.