1 - Functions and Modeling

Function Notation and Evaluation

A function describes a dependable input-output relationship: every permitted input has exactly one output. The input is often called the independent variable, and the output is often called the dependent variable.

If $f(x)=3x^2-2x+1$, evaluate the function by substituting the input everywhere the variable appears:

Function notation does not mean multiplication; $f(x)$ means the output of $f$ at input $x$. A relation such as

is a function because no input is paired with two different outputs. By contrast,

is not a function because the input $1$ has two outputs.

Takeaway: Check the input-output rule first, then use substitution to evaluate a function.

Domain and Range

The domain records which inputs are allowed, while the range records which outputs actually occur. Both mathematical rules and real-world context can restrict the possible values.

For a formula, inspect common sources of undefined expressions:

• A denominator cannot be zero. For $f(x)=\frac{x+2}{x-3}$, exclude $x=3$, giving the domain $(-\infty,3)\cup(3,\infty)$.

• The radicand of an even-index radical must be nonnegative. For $g(x)=\sqrt{x-5}$, require $x-5\ge 0$, so the domain is $[5,\infty)$.

• The argument of a logarithm must be positive. For $h(x)=\log(x+1)$, require $x+1>0$, so the domain is $(-1,\infty)$.

Context may add restrictions. Time may require $t\ge 0$, and a number of manufactured items may need to be a nonnegative integer. To find a range, inspect the formula, graph, table, or context. For $f(x)=(x-2)^2+3$, the square is never negative, so the range is $[3,\infty)$.

Takeaway: State both mathematical and contextual restrictions instead of assuming every real input is allowed.

Representations and Graph Behavior

A relationship can be expressed verbally, numerically, graphically, or algebraically. Moving between these forms helps reveal patterns and supports reasonable interpretation.

A table with inputs $0,1,2,3$ and outputs $2,5,8,11$ has a constant output increase of $3$ for every input increase of $1$. This supports the linear rule

A table alone may not determine a unique formula for inputs that were not observed, so predictions beyond the table require a justified model.

For graphs, the horizontal axis shows inputs and the vertical axis shows outputs. Use the Vertical line test to decide whether the graph represents a function. The domain is the set of horizontal coordinates reached, and the range is the set of vertical coordinates reached. Intercepts, increasing or decreasing intervals, maximums, minimums, and asymptotes describe behavior.

Takeaway: Use the representation that makes the question easiest, then check that all representations tell a consistent story.

Transformations of Functions

Transformations modify a parent graph without requiring a completely new graph from scratch. In

the changes can be read from the constants, but horizontal changes occur inside the function and therefore require careful attention to signs.

• $f(x)+k$ shifts the graph up when $k>0$ and down when $k<0$.

• $f(x-h)$ shifts the graph right by $h$ when $h>0$.

• $-f(x)$ reflects the graph across the horizontal axis.

• $f(-x)$ reflects the graph across the vertical axis.

• Multiplication by $a$ stretches vertically when $|a|>1$ and compresses vertically when $0<|a|<1$.

• Replacing $x$ by $bx$ compresses horizontally when $|b|>1$ and stretches horizontally when $0<|b|<1$.

For example,

is the graph of $f(x)=x^2$ shifted right $4$ units and up $2$ units, so its vertex is $(4,2)$.

For a periodic model,

$|A|$ is the amplitude, $h$ is the horizontal shift, $D$ gives the midline, and the period is $\frac{2\pi}{|B|}$ when angles are measured in radians.

Takeaway: Identify outside changes, inside changes, reflections, and scale factors separately before sketching the result.

Composition of Functions

Composition of Functions connects two processes by feeding the output of one function into another. In

the function on the right, $g$, is evaluated first.

Let $f(x)=2x+1$ and $g(x)=x^2$. Then

while

The results differ, so composition is generally not commutative.

The domain of a composition must satisfy both stages: the input must be in the domain of the inner function, and the inner output must be in the domain of the outer function. If $f(x)=\sqrt{x}$ and $g(x)=x-3$, then

which requires $x-3\ge 0$. Therefore, the domain is $[3,\infty)$.

Takeaway: Read a composition from the inside out and verify restrictions at both stages.

Inverse Functions

An Inverse Function reverses the original input-output process, so the original function must be one-to-one on the stated domain. The horizontal line test checks this graphically.

To find the inverse of $f(x)=3x-5$, write

Interchange the variables and solve for the new output:

Thus,

A check confirms the reversal:

The domain and range switch:

The graph of the inverse is the reflection of the original graph across $y=x$. For example, $f(x)=x^2$ is not one-to-one over all real numbers because inputs $2$ and $-2$ both produce $4$. Restricting its domain to $[0,\infty)$ makes an inverse function possible.

Takeaway: Check one-to-one behavior, exchange the variables, solve carefully, and verify by composition.

Modeling and Interpretation

Modeling begins by identifying what the variables mean and selecting a representation that captures the situation. A useful process is:

1. Identify the input and output variables.

2. State the domain, including contextual restrictions.

3. Choose a table, graph, formula, or verbal description.

4. Look for structure such as constant differences, ratios, turning points, asymptotes, or repeating cycles.

5. Interpret important features in context.

6. Check the model against known values and decide whether its predictions are reasonable.

For a cost model such as

$C(20)=390$. The coefficient $12$ represents the change in cost per unit, while $150$ is the initial value when $n=0$, provided that interpretation fits the context.

For repeating data such as ocean height, a sinusoidal model can describe recurring increases and decreases. Its midline represents the average level, its amplitude represents the maximum deviation from that level, and its period represents the time between corresponding points in successive cycles.

Takeaway: A model is useful when its algebraic behavior, graph, domain, and interpretation agree with the situation.

• One-to-one function

• Transformation of Functions