Problem 43 - College Algebra

Question

a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). $$ f(x)=(x-1)^{2}, x \leq 1 $$

Step-by-Step Solution

Answer

The inverse of the function \(f(x)=(x-1)^2\) is \(f^{-1}(x)=\sqrt{x} + 1\). The original function \(f(x)\) has a domain and range of \([-∞, 1]\) and \([0, ∞)\) respectively, while its inverse \(f^{-1}(x)\) has a domain and range of \([1, ∞)\) and \([1, ∞)\) respectively.

Step 1: Find the Inverse of the Function

To find the inverse of a function, we first replace the function variable \(f(x)\) with y, meaning we'll have \(y=(x-1)^2\). Then, swap x and y to get \(x=(y-1)^2\). After exchanging, solve for \(y\). This will get us \(y=\sqrt{x} + 1\) or \(y=-\sqrt{x} + 1\). Since x for \(f^{-1}(x)\) shouldn't be less than 1, we disregard \(y=-\sqrt{x} + 1\). So, the inverse of the function is \(f^{-1}(x)=\sqrt{x} + 1\).

Step 2: Plot the Function and Its Inverse

To draw the graph, first plot the original function \(f(x)=(x-1)^2\) where \(x \leq 1\). This will produce a parabola. For the inverse \(f^{-1}(x)=\sqrt{x} + 1\), expect a curve line branching out upwards, indicating all values of \(x\geq 1\). Plotting both functions on the same coordinate system will show that the two graphs are mirror images with respect to the line \(y=x\).

Step 3: Finding the Domain and Range

The domain and range of a function are the set of all possible input and output values of that function respectively. For \(f(x)\), the domain is \(x\leq 1\) or \([-∞, 1]\) in interval notation and the range is any \(y\) such that \(y \geq 0\) or \([0, ∞)\) in interval notation. For \(f^{-1}(x)\), it's the opposite; the domain is any \(x\) such that \(x\geq 1\) or \([1, ∞)\) and the range is any \(y\) such that \(y\geq 1\) or \([1, ∞)\).

Key Concepts

Finding the Inverse of a Function

Understanding how to find the inverse of a function unlocks the ability to uncover a mirror image relationship between two sets of data. To determine the inverse, follow these steps:

• Start with the original function, written as either f(x) or y. For instance, if we have f(x) = (x-1)^2, begin by writing y = (x-1)^2.
• Interchange the x and y in the equation to reflect their inverse relationship. The equation now appears as x = (y-1)^2.
• Solve the new equation for y. This step might involve algebraic manipulation, including undoing any operations that were done to y. In our example, by finding the square root of both sides, we get two solutions: y = sqrt(x) + 1 and y = -sqrt(x) + 1. However, the context or restrictions of the original function guide which solution to choose. Since the original function f(x) is defined for x ≤ 1, for its inverse we only consider the positive square root.
• The resulting expression is the inverse function, denoted f^-1(x).

It is essential to remember that not all functions have inverses, and inverses may only exist under certain conditions.

Graphing Functions and Inverses

The depiction of functions and their inverses on a graph facilitates a visual understanding of their relationship. When graphing, please consider:

• The graph of an inverse function is a reflection of the original function across the line y = x. If you fold the graph along this line, the function and its inverse should match up.
• To graph the original function f(x) = (x - 1)^2 for x ≤ 1, sketch a downward-opening parabola that stops at the vertex, located at the point (1,0), since x is restricted to 1 or less.
• Next, graph the inverse function f^-1(x) = sqrt(x) + 1. It starts at the point (1,1) and increases, with the curve proceeding in the opposite direction of the original function, but keeping a consistent reflection across the line y = x.
• Remember to always verify if the inverse correctly mirrors the original function by checking if the points reflecting through y = x lie on both diagrams.

Domain and Range in Interval Notation

Interval notation is a succinct way to express the domain (input values) and range (output values) of functions. It helps to:

• Identify the smallest and largest values of x for the domain, and y for the range. The notation uses brackets [ ] to indicate that the endpoint is included—an interval that contains its endpoint—and parentheses ( ) to show the endpoint is not included—an interval that approaches but never reaches its endpoint.
• For the original function f(x) = (x - 1)^2, the domain is all x less than or equal to 1, expressed as (-∞, 1], and the range is all y values greater than or equal to 0, denoted as [0, ∞).
• Conversely, for the inverse function f^-1(x), the domain and range swap roles. The domain is now [1, ∞), and the range becomes [1, ∞) too.
• Note that infinity is never included in the interval, hence the parentheses are used when referring to ∞ or -∞ regardless of the inequality sign.

By mastering interval notation, clarifying the scope of inputs and outputs for functions and their inverses becomes straightforward and precise.