Problem 76 - 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.) $$ g(x)=(x-1)^{2} $$

Step-by-Step Solution

Answer

Restrict domain to \(x \geq 1\); inverse is \(g^{-1}(x) = \sqrt{x} + 1\).

Step 1: Identify the One-to-One Property

A function is one-to-one if each y-value corresponds to exactly one x-value. The function \(g(x)=(x-1)^2\) is not one-to-one over its entire domain because it is a quadratic function with the vertex at \((1, 0)\), opening upwards and symmetric about the line \(x = 1\).

Step 2: Restrict the Domain

To make \(g(x) = (x-1)^2\) one-to-one, we choose a portion of the domain where the function is strictly increasing or decreasing. Let's restrict the domain to \(x \geq 1\). In this domain, the function \(g(x)\) increases monotonically.

Step 3: Find the Inverse Function

With the restricted domain \(x \geq 1\), solve \(y = (x-1)^2\) for \(x\). First, take the square root of both sides to obtain \(\sqrt{y} = x - 1\). Then, rearrange to solve for \(x\): \(x = \sqrt{y} + 1\).

Step 4: Write the Inverse Function

Since \(x = \sqrt{y} + 1\), the inverse function is \(g^{-1}(x) = \sqrt{x} + 1\). This inverse function corresponds to the function \(g(x) = (x-1)^2\) with the domain restricted to \(x \geq 1\).

Key Concepts

Quadratic Functions

Quadratic functions are mathematical expressions of the form \(f(x) = ax^2 + bx + c\). They are characterized by their U-shaped graphs, known as parabolas. These functions often appear in many areas of mathematics and real-world applications. The key features of quadratic functions include:

• A quadratic function has a vertical line of symmetry called the axis of symmetry, cutting the parabola into two mirror-image halves.
• It has a vertex, which is the highest or lowest point on the graph, depending on whether it opens upwards or downwards.
• The standard form allows you to identify the direction of the parabola. If \(a > 0\), it opens upwards; if \(a < 0\), it opens downwards.

Understanding these properties helps to manipulate quadratic functions effectively, such as when finding their inverses. This is crucial when modifying functions like \(g(x) = (x-1)^2\) to become one-to-one by restricting its domain.

Domain Restriction

Restricting the domain of a function is a technique used to make a non-one-to-one function, like a quadratic, one-to-one. A domain is the set of all possible input values (x-values) for a function. In our exercise, the function \(g(x) = (x-1)^2\) initially had a domain of all real numbers. To make \(g(x)\) one-to-one, we must restrict its domain.

By limiting the domain to \(x \geq 1\), we ensure the function becomes strictly increasing. This way, each y-value has only one corresponding x-value. This restriction is crucial for finding a valid inverse because only invertible functions can have well-defined inverses. It highlights the importance of domain analysis for quadratic functions, enabling us to manipulate them to meet specific requirements. Think of domain restriction as a way to "zoom in" on a portion of the function where it behaves predictably.

One-to-One Functions

A one-to-one function is a special kind of function where each output value is unique to one input value. This means no two different input values produce the same output. Such functions are vital for finding inverses. In mathematical terms, a function \(f(x)\) is one-to-one if, whenever \(f(x_1) = f(x_2)\), then \(x_1 = x_2\).

For the function \(g(x) = (x-1)^2\), simply modifying its domain from \([-\infty, \infty]\) to \([1, \infty)\) transforms it into a one-to-one function. This is crucial for finding the inverse. With \(g(x)\) now being one-to-one, each y-value corresponds to exactly one x-value, enabling us to find \(g^{-1}(x)\), the inverse function, which is \(g^{-1}(x) = \sqrt{x} + 1\).

Recognizing and transforming functions into a one-to-one form allows for deeper mathematical analysis and applications, such as solving problems that require reversing operations.