functions

Functions are mathematical constructs that describe relationships between quantities.
Various professionals analyze dependencies between quantities using functions.
- Manufacturers: Interested in profit as a function of production levels.
- Biologists: Analyze population changes of cultures (e.g., bacteria) over time.
- Psychologists: Study relationships between learning time and vocabulary list length.
- Chemists: Examine how reaction speed correlates with substrate amounts.

A function expresses a dependency of one variable on another.
Mathematically, a function is represented as $y = f(x)$ , where:
- y: Dependent variable (output)
- x: Independent variable (input)

Let:
- $x$ = radius of circle
- $y$ = area of circle
From geometry, the area is defined as:
- $y = ext{π} imes x^2$
This can also be expressed as:
- $f(x) = ext{π} imes x^2$
For a radius of 5 inches:
- $f(5) = ext{π} imes (5)^2 = 25 ext{π}$ square inches.

In a function $y = f(x)$ :
- Independent Variable: $x$
- Dependent Variable: $y$
Domain: Set of all possible $x$ values (input values).
Range: Set of all possible $y$ values (output values).

For the area function $f(x) = ext{π} imes x^2$ :
- Domain: All positive values of $x$ (radius cannot be negative).
- Range: All positive values of $y$ (area is always positive).

Linear Function Definition: A function that can be represented in the slope-intercept form:
- $y = mx + b$
- Here, $m$ is the slope and $b$ is the y-intercept.
This can also be expressed as:
- $f(x) = mx + b$
Importance: Linear functions are crucial for analyzing quantitative relationships in business and economics.

Trend Analysis: Rapid growth of the over-65 population projected to increase healthcare spending significantly in upcoming decades.
Mathematical Model: Health-care expenditures from 2008 through 2013 modeled as:
- $S(t) = 0.134t + 2.325$ ,
- where $t$ represents years since 2008 (i.e., $t = 0$ corresponds to 2008).

Graphing the Function: Sketch $S(t)$ alongside projected health-care expenditure data from 2008-2013.
Expenditure Projection for 2014:
- Calculate: $S(6) = 0.134 imes 6 + 2.325 = 3.129 ext{ trillion} <br>ightarrow ext{Approx. } 3.13 ext{ trillion}$ .
Rate of Increase: Derived from the slope:
- $m ext{(slope)} <br>ightarrow ext{Rate of increase} = 0.134 ext{ trillion/year} ext{ or approx. } 0.13 ext{ trillion/year}$ .

Depreciation describes the reduction in value of an asset over time.
Example Case: A network server with an original value of $10,000$ :
- Decreased linearly over 5 years to a scrap value of $3,000$ .

Let $V(t)$ represent the book value of the server at the end of year $t$ :
- Since depreciation is linear, $V$ is a linear function of $t$ .

Using known values:
- At $t = 0$ : $V(0) = 10,000$ (initial value).
- At $t = 5$ : $V(5) = 3,000$ (scrap value).
Calculate the slope (rate of depreciation):
- Using points: $V(0) = (0, 10000)$ and $V(5) = (5, 3000)$ .
- Slope $m$ can be found as:
- $m = \frac{V(5) - V(0)}{5 - 0} = \frac{3000 - 10000}{5 - 0} = -1400$ .

Use the point-slope form to derive the line equation:
- $V - 10,000 = -1400(t - 0)$
- Therefore: $V(t) = -1400t + 10,000$

Book Value at End of Year 2:
- $V(2) = -1400(2) + 10,000 = 7200$
Depreciation Rate:
- Rate is given by the slope's negative value:
- Since $m = -1400$ , the rate of depreciation is $1400 per year.

Graph of $V(t) = -1400t + 10000$ :
- Starting point: (0, 10000)
- Ending point: (5, 3000)
- Depicts linear depreciation from initial value to the scrap value over 5 years.