Problem 69 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine algebraically whether the function is one-to-one. Verify your answer graphically. If the function is one-to-one, find its inverse. $$f(x)=|x-2|, \quad x \leq 2$$

Step-by-Step Solution

Answer

The function \(f(x)\) is one-to-one over the interval \(x \leq 2\) and its inverse function is \(f^{-1}(x) = 2 - x\).

Step 1: Check if the function is one-to-one

Since we are given a specific domain, \(x \leq 2\), consider the piecewise function: \(f(x) = 2-x\) if \(x \leq 2\). Note that the absolute value function \(|x-2|\) is equivalent to \(2-x\) in the interval \(x \leq 2\). Now, for this piece of function to be one-to-one, each x-value in the domain must correspond to exactly one y-value. Since \(f(x)\) is a linear function over this interval with a slope of -1, it is indeed one-to-one as it will pass both vertical and horizontal line tests.

Step 2: Find the inverse of the function

The inverse of a function 'undoes' the operation of the original function. The inverse of \(f(x)\) is denoted by \(f^{-1}(x)\). If \(y=f(x)\), then \(x=f^{-1}(y)\). Using this principle, if \(y = 2 - x\), the inverse function can be found by switching x and y, resulting in \(x = 2 - y\). Solving this for y, we get the inverse function \(f^{-1}(x) = 2 - x\).

Key Concepts

One-to-One Functions

A one-to-one function is a special type of function where each input is associated with a unique output. This means that no two different inputs map to the same output. In simpler terms, if you think of a list of people and their favorite foods, no single food can be the favorite of more than one person.

Mathematically, a function is considered one-to-one if for any two distinct inputs, say \(x_1\) and \(x_2\), the outputs \(f(x_1)\) and \(f(x_2)\) are different. A simple way to test for one-to-one functions is to use the horizontal line test.

• If a horizontal line drawn through any part of the graph of the function intersects it at more than one point, the function is not one-to-one.
• If it intersects at only one point, the function is one-to-one.

Using the horizontal line test on the given piece of \(f(x) = |x - 2|, x \leq 2\), we observe that a horizontal line will never intersect the linear section \(f(x) = 2 - x\) more than once. Thus, it is one-to-one within its domain.

Piecewise Functions

Piecewise functions are functions that are defined by different expressions over different intervals of their domain. They are like chameleons, changeable according to the section of the number line they inhabit.

In this case, the function \(f(x) = |x - 2|\) was simplified to a piecewise linear function \(f(x) = 2 - x\) for \(x \leq 2\). This happens because the absolute value affects the output differently based on the input's relation to 2. For values less or equal to 2, the function simplifies to \(2-x\).

• This creation of distinct function "pieces" helps to adapt the function to various conditions found in the domain.
• Understanding each piece separately is crucial when analyzing behavior like one-to-one properties or finding inverses.

Vertical and Horizontal Line Tests

The vertical and horizontal line tests are graphical tools used to determine the nature of a function.

The vertical line test checks if a graph represents a function. Draw a vertical line anywhere on the graph.

• If it intersects the graph at more than one point, it's not a function.
• If it intersects at exactly one point at every location, it is a function.

The horizontal line test, on the other hand, is used to determine if a function is one-to-one. This involves drawing horizontal lines across the graph.

• If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
• If it never does, the function is one-to-one.

Using these tests on \(f(x) = 2-x\) over its defined interval \(x \leq 2\), confirm that it is a function by vertical line test and is one-to-one by horizontal line test, since each line intersects the function at most once.