Question

Functions That Agree at All but One Point (a) In the context of finding limits, discuss what is meant by two functions that agree at all but one point. (b) Give an example of two functions that agree at all but one point.

Short answer

When two functions agree on all but one point, it means that for all inputs, the two functions share the same output, except for a specific input where they differ. As an example, consider the functions \(f(x) = x\) and \( g(x)\) where \(g(x) = x\) for all \(x \neq 1\), and \(g(x) = 2\) when \(x = 1\). These functions agree on all points except at x=1.

Step-by-step solution

Explaining the Concept

Two functions that agree at all but one point means that for every input in their domain, the output or the function values are similar, except for one specific input, where the function values may differ.

Provide an Example

An example can consist of two simple mathematical functions: \(f(x) = x\) and \(g(x) = \begin{cases} x, & \text{if } x \neq 1 \\ 2, & \text{if } x = 1 \end{cases}\). The two functions have the same value for all \(x\), except at \(x = 1\). At this point, while \(f(1) = 1\), the function value of \(g\) is 2.

Key concepts

Functions

Functions are fundamental concepts in mathematics that describe relationships between sets. In simple terms, a function is a rule that assigns every input exactly one output. Imagine a vending machine: when you press a button (input), you get a specific drink (output). Here, the vending machine acts like a function.
In mathematical terms, if we have a set of inputs called the domain and a set of possible outputs called the codomain, a function will map each element from the domain to an element in the codomain. The notation \( f(x) \) is commonly used, where \( x \) represents an input and \( f(x) \) is the corresponding output. For example, with a function \( f(x) = x^2 \), the input \( 3 \) would yield an output of \( 9 \).
Functions can be represented in various formats: equations, graphs, tables, and even words. What’s important is that for every input, there's a unique output. This characteristic distinguishes functions from other types of relations.

Function Domain

The function domain is a set of all possible inputs for which a function is defined. Think of the domain as the collection of all ingredients that go into our function. Not every input makes sense for every function, just like not every ingredient is suitable for every recipe.
For example, consider the function \( f(x) = \frac{1}{x} \). This function is undefined when \( x = 0 \), because division by zero is not possible in ordinary arithmetic. Thus, the domain of \( f(x) \) excludes \( 0 \) and consists of all other real numbers. You can also encounter functions with restricted domains due to square roots, logarithms, and other operations that may limit the possible inputs.
In practice, finding the domain requires checking these limitations and ensuring the inputs keep within the rules of algebra. It is critical in understanding the scope of what the function can operate on.

Piecewise Functions

Piecewise functions are a unique type of function defined by different expressions based on the input’s value. They are like a custom-built toolkit where each tool (piece of the function) serves a specific purpose.
For example, consider \( g(x) = \begin{cases} x, & \text{if } x eq 1 \ 2, & \text{if } x = 1 \end{cases} \). This is a piecewise function because it applies different expressions depending on whether \( x \) is equal to 1 or not. For most \( x \), \( g(x) = x \), but at \( x = 1 \), the function "switches gears" and outputs 2 instead.
Piecewise functions are super handy for modeling situations where a rule changes based on circumstances. They pop up in real-world scenarios such as tax brackets, delivery charges, or any process where conditions affect outcomes. Understanding how to read and construct these functions takes practice but equips you with powerful tools for tackling complex problems.