Problem 74 - Algebra and Trigonometry

Question

A one-to-one function is given. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line \(y=x .\) $$ g(x)=x^{2}+1, \quad x \geq 0 $$

Step-by-Step Solution

Answer

The inverse is \( g^{-1}(x) = \sqrt{x-1} \). Verifying graphs should show they reflect over \( y = x \).

Step 1: Identify the Function Limits

We are given the function \( g(x) = x^2 + 1 \) with the condition \( x \geq 0 \). This condition ensures the function is one-to-one and thus has an inverse.

Step 2: Set Up Equation for Inverse

To find the inverse function, set \( y = g(x) \), so \( y = x^2 + 1 \). Now, solve for \( x \) in terms of \( y \).

Step 3: Rearrange Equation

Set the equation \( y = x^2 + 1 \) to solve for \( x \). Subtract 1 from both sides to get \( y - 1 = x^2 \).

Step 4: Solve for \( x \)

Take the square root of both sides to solve for \( x \). Since \( x \geq 0 \), we only consider the positive square root: \( x = \sqrt{y-1} \).

Step 5: Write the Inverse Function

Since \( g(x) \) pairs \( x \) with \( y \), its inverse \( g^{-1}(y) \) pairs \( y \) with \( x \). Therefore, the inverse function is \( g^{-1}(x) = \sqrt{x-1} \).

Step 6: Graph the Functions

Graph \( g(x) = x^2 + 1 \) and its inverse \( g^{-1}(x) = \sqrt{x-1} \) on the same axes. Also, graph the line \( y=x \) to verify the functions are reflections of each other.

Key Concepts

One-to-One Functions

Understanding one-to-one functions is essential when dealing with inverse functions. A one-to-one function pairs each element in the domain with a unique element in the range. This characteristic ensures that the function has an inverse since no more than one output corresponds to the same input.
This uniqueness allows us to reverse the process and directly find an input from an output, which is the core idea behind finding an inverse function.
For a function to be one-to-one, it must pass the horizontal line test. This means that no horizontal line can intersect the graph of the function more than once. In the given exercise, the function \(g(x) = x^2 + 1\) with the condition \(x \geq 0\), is one-to-one because the restriction on the domain eliminates the part where the graph fails the horizontal line test.
This restriction makes it possible to find an inverse function for \(g(x)\).

Graphing Functions

Graphing functions is a crucial skill and helps visualize how a function behaves across its domain. For equations, a graph provides a picture of the function's growth, decline, symmetry, and intercepts. In the exercise given, after establishing that \(g(x) = x^2 + 1\) is one-to-one for \(x \geq 0\), we can graph this alongside its inverse.
When graphing both a function and its inverse, it is important to include the line \(y = x\). This line serves as the reflection mirror. If the functions are graphed correctly, the function and its inverse should appear as mirror images over the line \(y = x\).
This reflection property is an important check. It's a visual confirmation that our inverse function, \(g^{-1}(x) = \sqrt{x-1}\), is correct, and our understanding and application of inverse functions are solid. The graphs not only show symmetry but also help identify potential domain and range issues before solving them algebraically.

Function Limits

Function limits are restrictions placed on the domain or range of a function. These restrictions ensure that conditions such as one-to-one-ness are met, which are necessary to find an inverse.
In our scenario, the function \(g(x) = x^2 + 1\) is only considered for \(x \geq 0\). This restriction turns a naturally not one-to-one function (a quadratic) into a one-to-one function by removing the left-side of its parabolic curve. Function limits are not just arbitrary conditions but are critical to ensuring the mathematical properties required for finding inverses or solving equations are met.
By limiting the domain to \(x \geq 0\), we clear the way for finding \(g^{-1}(x)\) without any ambiguity. This also alters the range of our function and its inverse, which must be understood and respected while solving or graphing.