01 Functions and Mathematical Models

Functions as Models

A Function assigns exactly one output to each permitted input. It can be written as $y=f(x)$, where $x$ is the independent variable and $y$, or $f(x)$, is the dependent variable. A graph represents a function when every vertical line intersects it at most once; this is the vertical line test.

A taxi that charges a fixed fee of $\$4$ plus $\$2.50$ per mile can be represented by

Here, $m$ is measured in miles, $C(m)$ is measured in dollars, $C(0)=4$ is the initial value, and $2.50$ is the rate of change in dollars per mile. This is a Mathematical model because it translates a situation into a mathematical rule.

A model is useful only when its variables, units, assumptions, and allowed inputs match the situation. A formula may be defined for every real input but still be meaningful only for nonnegative time or whole-number quantities.

Takeaway: A function is a rule connecting inputs and outputs, and a model adds the context needed to interpret that rule.

Domain, Range, and Restrictions

The Domain is the set of allowable inputs, and the Range is the set of outputs generated by those inputs. If no domain is specified, use all real inputs for which the formula is defined, then apply any practical restrictions from the situation.

For real-valued functions, check these restrictions:

• A denominator cannot equal zero.

• The radicand of an even root must be nonnegative.

• The argument of a logarithm must be positive.

• A trigonometric expression must be defined; for example, $\tan x$ is undefined when $\cos x=0$.

Examples:

• The polynomial $f(x)=x^3-4x+1$ has domain $(- \infty,\infty)$.

• The rational function $g(x)=\frac{x+1}{x-3}$ has domain $(- \infty,3)\cup(3,\infty)$.

• For $h(x)=\sqrt{5-2x}$, the condition $5-2x\ge0$ gives domain $(- \infty,\frac52]$ and range $[0,\infty)$.

• For $p(x)=\ln(x-4)$, the condition $x-4>0$ gives domain $(4,\infty)$ and range $(- \infty,\infty)$.

On a graph, the domain is the horizontal extent and the range is the vertical extent. Filled points generally include endpoints, open circles exclude them, and arrows show continuation. A graph can also reveal intercepts, extrema, asymptotes, and intervals of increase or decrease.

Takeaway: Determine algebraic restrictions first, then check whether the context narrows the domain further.

Transforming Parent Functions

A Transformation relates a complicated graph to a simpler parent function. Starting with $y=f(x)$, consider

The parameters have these effects:

• $a$ controls vertical stretch or compression; if $a<0$, it reflects the graph across the $x$-axis.

• $b$ controls horizontal scaling; if $b<0$, it reflects the graph across the $y$-axis.

• $h$ shifts the graph horizontally by $h$ units.

• $k$ shifts the graph vertically by $k$ units.

The horizontal scale factor is $1/|b|$. Thus, a larger value of $|b|$ produces a horizontal compression. Read horizontal changes from the entire expression inside the function.

For example,

is the graph of $x^2$ shifted right $3$ units and up $2$ units. Also,

is $\sqrt{x}$ shifted left $4$ units and down $1$ unit. Its domain changes from $[0,\infty)$ to $[-4,\infty)$, and its range changes from $[0,\infty)$ to $[-1,\infty)$.

Takeaway: Transformations change the graph and may also change its domain and range.

Composition and Inverses

A Composition applies one function after another:

The inside function, $g$, acts first; its output then becomes the input of $f$. For example, let

Then

whereas

Because these results differ, $f\circ g\ne g\circ f$. A composition is defined only where the output of the first function lies in the domain of the second function.

An Inverse function reverses the input-output process. If $f(a)=b$, then $f^{-1}(b)=a$. The domain of the inverse is the range of the original function, and the range of the inverse is the domain of the original function. An inverse exists as a function only when the original function is one-to-one, which can be checked with the horizontal line test.

For $f(x)=3x-4$, solve $y=3x-4$ for $x$, obtaining $x=\frac{y+4}{3}$, and interchange the variables:

The graphs of a function and its inverse reflect across $y=x$. The notation $f^{-1}(x)$ means an inverse function, not the reciprocal $1/f(x)$.

Takeaway: Composition builds a multistage process, while an inverse undoes a one-to-one process.

Exponential and Logarithmic Models

An Exponential function has the form

where $a\ne0$, $b>0$, and $b\ne1$. Its domain is all real numbers. If $a>0$, its range is $(0,\infty)$; if $a<0$, its range is $(- \infty,0)$. The horizontal asymptote is $y=0$, and the $y$-intercept is $(0,a)$.

• When $b>1$, the function models exponential growth.

• When $0<b<1$, the function models exponential decay.

A discrete growth or decay model is

where $P_0$ is the initial amount. For a percentage growth rate $r$, use $b=1+r$. For a percentage decay rate $r$, use $b=1-r$. Continuous growth or decay is commonly modeled by

where $k>0$ indicates growth and $k<0$ indicates decay.

A Logarithmic function reverses an exponential relationship:

Its domain is $(0,\infty)$, its range is all real numbers, and its vertical asymptote is $x=0$. Useful properties include

and the change-of-base formula

For example, solving $500=100(1.08)^t$ gives

Takeaway: Exponential functions model multiplicative change; logarithms reverse exponentiation and isolate unknown exponents.

Periodic and Trigonometric Models

A Periodic function repeats its values at regular intervals. The basic trigonometric functions are $\sin x$, $\cos x$, and $\tan x$. For $y=\sin x$ and $y=\cos x$, the domain is all real numbers, the range is $[-1,1]$, the period is $2\pi$, and the amplitude is $1$.

The tangent function is undefined when

because $\tan x=\frac{\sin x}{\cos x}$ and $\cos x=0$ at those inputs. Periodic functions are useful for modeling tides, seasonal variation, circular motion, sound waves, and other repeated phenomena.

A sinusoidal model can be written as

Its features are:

• amplitude: $|A|$;

• period: $\frac{2\pi}{|B|}$;

• horizontal shift: $h$;

• midline: $y=D$;

• maximum: $D+|A|$;

• minimum: $D-|A|$.

For

the midline is $12$, the amplitude is $3$, the maximum is $15$, the minimum is $9$, and the period is

If $t$ is measured in months, the model describes a cycle that repeats every $12$ months.

Takeaway: Identify amplitude, period, midline, and shifts from the parameters before interpreting a repeating pattern.

Interpreting Models in Context

A formula and its graph provide complementary information. Analyze a model in the following order:

1. Identify the variables and their units.

2. Determine the algebraic and contextual domain.

3. Find intercepts, endpoints, and convenient points.

4. Describe increases, decreases, extrema, asymptotes, periodicity, and end behavior.

5. Connect parameters to meanings such as rate, initial value, amplitude, or maximum deviation.

6. Check whether the predictions are reasonable in the situation.

Consider the height model

where $t$ is time in seconds and $h(t)$ is height in feet. The value $h(0)=5$ means the object starts $5$ feet above the ground. Because the coefficient of $t^2$ is negative, the graph opens downward, so its vertex gives the maximum height. The positive solution of $h(t)=0$ gives the time when the object reaches the ground.

Although this polynomial is defined for every real number, its contextual domain begins at $t=0$ and ends when the object reaches the ground. Negative time and values after impact are not meaningful for this model.

This distinction between the mathematical domain of a formula and the contextual domain of a model is essential. A mathematically valid expression does not automatically produce a meaningful prediction outside the assumptions of the situation.

Takeaway: Interpret every calculation using units, assumptions, and the meaningful domain.