# Analyzing Functions in Context

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## Representing a Function

## Independent and Dependent Variables

For functions in context, the input is often referred to as the independent variable

because it can be chosen arbitrarily from the domain. Conversely, the output is called the dependent variable,

because its value depends on the value of the independent variable. For example, consider how the cost of fruit is determined by weight.
$\begin{array}{c c c c c}
\text{cost} & = & \text{unit price } & \times & \text{ weight}\\
y & = & 2.5 & \cdot & x
\end{array}$

*depends*on how many pounds are bought. Therefore, the cost $y$ of clementines is the

**dependent variable**and the number of pounds $x$ is the

**independent variable**.

## Discrete Quantity

A discrete quantity is a quantity that can only take specific values. An example would be the number of times a person has gone skydiving. Notice that the value of that quantity can only be specific values like $\{0, 1, 2, 3, \ldots \}.$ If the independent variable of a function is a discrete quantity, it is said that the function has a *discrete domain*. Such a function can be recognized by its graph, which consists of any number of unconnected points.

## Continuous Quantity

A continuous quantity is a quantity that can take **any** value within one or several intervals. In other words, continuous quantities can be measured to any arbitrarily high degree of precision. Consider a person's age.

The age of a person is changing constantly, depending on how precisely it is measured. Counting only in years, a person could be $5$ or $6$ years old. Counting more precisely, he or she could be $16$ and a half, $10$ years and $3$ months, or even $25$ years and $172$ days old.

### Continuous Domain

If the input of a function is a continuous quantity, the function is said to have a *continuous domain*. The graph of such a function is typically a curve or a line.

## Interpreting Functions in Context

Functions in context can be interpreted in terms of domain and range, intercepts, and specific function values.

### Continuous and Discrete Range

Continuous and discrete range are defined similarly to continuous and discrete domain. If the dependent variable of a function is a discrete quantity, then it has a discrete range. If the dependent variable instead is a continuous quantity, then the function has a continuous range.

### Intercepts

The intercepts of a function in context are often of interest. For a function describing a cost, the $y$-intercept typically represents a starting cost, which does not depend on how much the service is used. For a function describing the altitude of an object, the $x$-intercept most often represents when or where the object touches the ground.

### Function Values

When a function value is given in the form $f(a) = b,$ for a function $f$ in context, there is always some interpretation that can be done. An example could be the function $C(t) = 3 + 0.2t,$

that describes the cost, $C,$ in dollars for jumping in a bouncy castle for $t$ minutes. The statement $C(25) = 8$ could then be interpreted as $``$jumping in the bouncy castle for $25$ minutes costs $8$ dollars." Note that this corresponds to the point $(25, 8)$ on the function graph.## Exercises

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