Consider the function $$N(t) = 110 + 25t$$, which describes how a baseline of $$110$$ unread emails increases by $$25$$ emails per day. In this relationship, the number of unread emails, $$N$$, is contingent upon the number of days, $$t$$. Consequently, the variable $$N$$ is the dependent variable, whereas the variable $$t$$ is the independent variable.

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For a mathematical function expressed in the notation $$y = f(x)$$, the output variable $$y$$ (which is equivalently written as $$f(x)$$) is known as the dependent variable. It is referred to as 'dependent' because its specific value relies entirely on the value chosen for the independent variable $$x$$.

Dependent Variable of a Mathematical Function

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For the function $$N(t) = 75 + 10t$$, which models an account starting with $$75$$ unread emails and growing by $$10$$ emails per day, the number of unread emails is a function of the number of days. The number of unread emails, $$N$$, depends on the number of days, $$t$$. Therefore, the variable $$N$$ acts as the dependent variable, while the variable $$t$$ serves as the independent variable.

Example: Determining the Independent and Dependent Variables for the Function $$N(t) = 75 + 10t$$

In the function $$N(t) = 100 + 15t$$, modeling an account that initially has $$100$$ unread emails and accumulates $$15$$ more each day, the total number of unread emails ($$N$$) is determined by the passage of time in days ($$t$$). Since the number of emails relies on the elapsed days, $$N$$ represents the dependent variable and $$t$$ represents the independent variable.

Example: Determining the Independent and Dependent Variables for the Function $$N(t) = 100 + 15t$$

Example: Determining the Independent and Dependent Variables for the Function $$N(t) = 110 + 25t$$

Evaluating a mathematical function is the process of finding the numerical value of $$f(x)$$ for a specific given value of the independent variable $$x$$. To perform this process, substitute the provided value for $$x$$ into the function's equation, and then simplify the resulting numerical expression by following the order of operations.

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