Problem 53 - Precalculus : Building Concepts and Connections

Question

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=(x-1)^{2}, x \geq 1$$

Step-by-Step Solution

Answer

The inverse function of \( g(x) = (x - 1)^{2} \), \( x \geq 1 \), is \( g^{-1}(x) = \sqrt{x} + 1 \), \( y \geq 1 \). The graph of both functions consists of the parabola \( g(x) \) in the first quadrant and the curve of \( g^{-1}(x) \), a reflection of the original function across the line \( y = x \).

Step 1: Find the Inverse of the Function

Denote the inverse function as \( g^{-1}(x) \). Swap the roles of \( x \) and \( y \) and solve for \( y \) in the equation \( x = (y - 1)^2 \) to find the inverse function. This equates to \( y = \sqrt{x} + 1 \). However, since the original function was only defined for \( x \geq 1 \), the inverse function, \( g^{-1}(x) \) will be \( y = \sqrt{x} + 1 \), for \( y \geq 1 \).

Step 2: Find the Domain of the Inverse Function

In the original function, \( x \geq 1 \). This means that in the inverse, \( y \geq 1 \). The domain of \( g^{-1}(x) \) is \([1, \infty)\).

Step 3: Graph the Functions

First, plot the function \( g(x) = (x - 1)^2 \), for \( x \geq 1 \), then plot the inverse, \( g^{-1}(x) = \sqrt{x} + 1 \), for \( y \geq 1 \). The two curves should be reflections of each other across the line \( y = x \).

Key Concepts

Graphing Functions

Understanding how to graph functions is essential for visualizing the relationship between variables in an equation. The process typically involves identifying points that satisfy the function and then connecting these points to reveal the shape on a set of axes.

For the function given in the exercise, \( g(x) = (x - 1)^{2} \), for \( x \geq 1 \), you start by choosing values of \( x \) within the domain \( [1, \infty) \) and calculate the corresponding \( y \) values. For example, if \( x = 1 \), then \( y = (1-1)^{2} = 0 \). Repeat for other values of \( x \), say 2, 3, 4, and so on, plotting the resulting points. The graph of \( g(x) \) is a parabola opening upwards, shifted one unit to the right of the y-axis.

When graphing on paper or digitally, use a consistent scale and clearly mark the axes. Connect the points with a smooth curve and label the function. For digital platforms, where learners interact with graphing, enabling features such as 'zoom' and 'pan' can significantly enhance the user experience.

Function Inverses

The inverse of a function, often denoted as \( f^{-1}(x) \), is a function that 'undoes' the effect of the original function. To find an inverse, we essentially reverse the roles of \( x \) and \( y \) and solve for \( y \). The key note here is that a function must be a one-to-one correspondence to have an inverse that is also a function.

In the given exercise, swapping \( x \) and \( y \) in the equation \( x = (y - 1)^{2} \) and then solving for \( y \) gives the inverse function. However, it is important to consider any restrictions on the domain or range as these will affect the inverse. Here, since the original function was defined only for \( x \geq 1 \), we take the positive square root, which guarantees \( y \geq 1 \) for the inverse function.

Remember, the graph of an inverse function is the reflection of the original function's graph across the line \( y = x \). Providing a visual example of this reflection on an interactive graph tool can greatly help students understand the concept of function inverses.

Domain of a Function

The domain of a function encompasses all the possible input values (\( x \)-values) for which the function is defined and yields real number outputs. For instance, in the original function \( g(x) = (x - 1)^{2} \), the domain is restricted to \( x \geq 1 \), as stated in the exercise. This is due to the function being defined only for those values.

When finding the domain of the inverse function, keep in mind that it corresponds to the range of the original function. Therefore, in the inverse function \( g^{-1}(x) = \sqrt{x} + 1 \), the domain reflects the condition that \( y \geq 1 \), so it is \( [1, \infty) \). Understanding the relationship between the domain and range, as well as how they interchange when computing inverses, is pivotal.

Clear and interactive examples, allowing students to manipulate the function and observe changes in domain and range, are excellent ways to reinforce these concepts. Moreover, visual cues such as highlighting the domain on the graph can aid in comprehension.