Question

Give an example of two functions that agree at all but one point.

Short answer

An example of two functions that agree at all but one point are \( f(x) = x \) and \( g(x) = x \) when \( x \neq 0 \), and \( g(x) = 1 \) when \( x = 0 \). These two functions agree at all x-values except for \( x = 0 \).

Step-by-step solution

Choose a Basic Function

Start by choosing a basic function that is easy to manipulate. An example could be a linear function. Let's take \(f(x) = x\). This function represents a straight line passing through the origin.

Identify the Point of Disagreement

Now, decide on a specific point on the graph of the function where the two functions will differ. In this case, let's choose \(x = 0\). For all other \(x\) values, the two functions will agree.

Define the Second Function

Define the second function in such a way that it agrees with the first function for all points except for the selected point. In this case, a good choice could be defining the second function to be \(g(x) = x\) when \(x \neq 0\) and \(g(x) = 1\) when \(x = 0\). Now, it is evident that for any x-value other than 0, f(x) = g(x); but at \(x = 0\), f(x) ≠ g(x) as required.