2 Functions as Mathematical Models

A progressive guide to modeling situations with functions, recognizing common function families, analyzing domains and ranges, interpreting transformations, and evaluating model assumptions in context.

Building Function Models

A function assigns exactly one output to each permitted input. A uses this structure to represent a real situation with variables, equations, tables, graphs, or functions.

To build a model, identify the quantities that change and the quantities being measured. Choose an input variable, often $x$ , and an output such as $y=f(x)$ . Then determine the relationship, state the , and interpret the constants with units.

For a taxi with a 3-unit starting fee and a 2-unit charge per mile, a reasonable model is

C(m)=3+2m.

Here, $m$ is the distance traveled and $C(m)$ is the cost. The practical begins at $m=0$ , because a negative distance is not meaningful. The equation could be evaluated at other numbers, but those values would not necessarily describe the situation.

Takeaway: A useful function model connects inputs and outputs while respecting the context, units, assumptions, and .

Choosing a Function Family

The pattern of change in data helps identify an appropriate function family.

A , $f(x)=mx+b$ , models constant additive change. The $m$ is the constant rate of change, and the $b$ is the output when $x=0$ . For $P(t)=120+15t$ , the initial value is 120 and the output increases by 15 units per time unit.
A polynomial function models smooth curves. A such as $f(x)=a(x-h)^2+k$ has a vertex at $(h,k)$ , while higher-degree polynomials can represent more changing behavior.
An , $f(x)=ab^x$ , models constant multiplicative or percentage change. If $b>1$ , it represents growth; if $0<b<1$ , it represents decay. In $A(t)=500(1.08)^t$ , the initial value is 500 and the factor 1.08 represents an 8 percent increase per time period.
A is the inverse of an . It is useful for quantities that increase rapidly at first and then more slowly. For $f(x)=\log_{10}(x)$ , multiplying the input by 10 increases the output by 1.
A uses different formulas on different intervals, such as a parking fee with one rule for the first two hours and another rule afterward.

Takeaway: Constant additive change suggests a linear model, constant multiplicative change suggests an exponential model, slowing growth suggests a logarithmic model, smooth turning suggests a polynomial model, and interval-dependent rules suggest a piecewise model.

Domains, Ranges, and Graphs

The contains allowable inputs, and the contains resulting outputs. Restrictions can come from algebra or from the real-world setting.

Common algebraic restrictions include the following:

A denominator cannot equal zero.
The expression inside an even root must be nonnegative for real-valued outputs.
The argument of a logarithm must be positive.

Context can impose additional restrictions. Counts of people or objects are usually not negative, and time in an application often begins at $t=0$ . For example, for

f(x)=\sqrt{x-4},

requiring $x-4\ge 0$ gives the $[4,\infty)$ . Because a square root is nonnegative, the is $[0,\infty)$ .

A graph shows the through its horizontal coordinates and the through its vertical coordinates. The checks whether a graph is a function: every vertical line must intersect the graph at no more than one point.

Takeaway: Determine restrictions before evaluating a function, and use both algebraic conditions and contextual meaning when stating the .

Transformations and Key Features

Transformations describe how a parent graph changes. A useful general form is

g(x)=a\,f\bigl(b(x-h)\bigr)+k.

The parameters have these effects:

$h$ shifts the graph horizontally; positive $h$ moves it right.
$k$ shifts the graph vertically; positive $k$ moves it up.
$|a|$ produces a vertical stretch or compression, and $a<0$ reflects the graph across the horizontal axis.
$|b|$ produces a horizontal compression or stretch, and $b<0$ reflects the graph across the vertical axis.

For

g(x)=-2(x-3)^2+5,

the parabola from $f(x)=x^2$ shifts right by 3 units, reflects across the horizontal axis, stretches vertically by a factor of 2, and shifts up by 5 units. Its vertex is $(3,5)$ , and because it opens downward, its is

(-\infty,5].

For a logarithmic example,

g(x)=\log_2(x+4)-1,

the graph shifts left by 4 units and down by 1 unit. The condition $x+4>0$ gives the $(-4,\infty)$ , and the vertical asymptote is $x=-4$ .

Takeaway: Read shifts, reflections, and stretches from the parameters, but check the function’s and after applying the .

Interpreting and Evaluating Models

Parameters acquire meaning from the situation in which a model is used. Consider

T(d)=72-0.5d,

where $T$ is temperature in degrees Celsius and $d$ is depth in meters. The 72 is the modeled temperature at depth 0. The $-0.5$ means that temperature decreases by 0.5 degrees Celsius per meter. If the model applies only from 0 to 40 meters, its meaningful is $0\le d\le 40$ .

The predicted value at 20 meters is

T(20)=72-0.5(20)=62.

A model should not automatically be trusted outside the where its assumptions are reasonable. Extrapolating far beyond observed or relevant inputs can produce misleading predictions. Compare predictions with observed values using graphs, residuals, or numerical error measures when available. A model may be mathematically convenient yet unsuitable if its , units, or assumptions do not fit the situation.

Takeaway: Interpret every parameter with units and context, and limit predictions to the where the model is justified.