91Ó°ÊÓ

A function is a special type of relation between two sets of values. In the context of math, we often deal with functions of a variable, where one variable (usually symbolized as \(x\)) is mapped to another variable (typically represented as \(y\)). For a relation to qualify as a function, each input value must correspond to exactly one output value.

Using our example, the relationship \(y = \frac{10}{x+2}\) describes \(y\) as a function of \(x\), provided each \(x\) value corresponds uniquely to a \(y\) value. In simpler terms, for every number you plug in for \(x\), you should get only one result for \(y\).

But what happens if a particular \(x\) value gives more than one result for \(y\)? Then, it wouldn't be a function. In the given exercise, once we exclude problematic \(x\) values, each remaining \(x\) gives out a unique \(y\), hence making our relation a function.

The concept of division by zero is crucial in mathematics. Division by zero doesn't just cause computational errors; it's actually undefined in mathematical terms. Think of division by zero as trying to divide something into water—it's impossible to see the resulting parts.

In any fraction like \(\frac{10}{x+2}\), the part \(x+2\) is the denominator, and it must never be zero. If it's zero, the division becomes undefined. Let's break this down to understand why:

• When \(x = -2\), adding 2 gives \(0\).
• This makes the denominator zero, resulting in \(\frac{10}{0}\).

Since this outcome is undefined, mathematicians agree that divisions involving zero in the denominator must be avoided.

The domain of a function is all the possible input values (values of \(x\)) that the function can accept. Some values might cause the function to be undefined, like triggering division by zero.

The exercise asks us to find those potential \(x\) values. Here's how we do it:

• Identify any \(x\) values that make the denominator zero.
• In our example, when \(x + 2 = 0\), \(x = -2\).

This means \(x = -2\) must be excluded from the domain since it makes the function undefined.

By excluding \(x = -2\) and using every other real number as inputs, we ensure our function works smoothly. This method of excluding certain values helps define the domain accurately, ensuring there are no undefined operations involved.