1.3 Functions - Comprehensive Notes

1.3 Functions

What is a function? (basic ideas)

• A function is a rule that assigns to each input value a single output value. It answers the question: given x, what is y?

• Example of a function: the rule y = f(x) = 2x + 1 is a function.

• Any line is a linear function; graphs of functions are curves (or lines) on the xy-plane.

• For any value of x, the rule must give at most one value of y. Therefore, vertical lines are not functions (a vertical line would give infinitely many y-values for a single x).

• Another classic example of a function is the parabola y = f(x) = x^2, which can be plotted by choosing x-values and plotting the corresponding (x, x^2) points, then drawing a smooth curve through them.

• Some functions can be evaluated for all real x (domain is all of (\mathbb{R})); others are only defined on certain intervals or subsets of (\mathbb{R}).

Domain and range concepts

• The domain of a function f is the set of all input values x for which f(x) is defined (the values you are allowed to plug in).

• The domain can be all real numbers, an interval, or a union of intervals; it is often written in interval notation.

• The endpoint conventions: in interval notation, square brackets [ ] mean the endpoint is included; parentheses ( ) mean the endpoint is not included.

• Examples of common domains:

  • The square-root function: y = f(x) = \sqrt{x} has domain x \ge 0\,'' i.e. ([0, \infty)).

  • The reciprocal function: y = f(x) = \frac{1}{x} has domain x \neq 0, i.e. all real numbers except 0, with a vertical asymptote at x = 0.

  • The quadratic function y = f(x) = x^2 has domain (-\infty, \infty)\,.

• The domain is often constrained by practical or story-context considerations (see below). The domain is what makes sense for the situation, not just what the symbols allow algebraically.

• In a graph, the domain corresponds to the set of x-coordinates for which there is a point on the graph; the range corresponds to the set of y-coordinates attained.

Representations of functions

• Functions can be defined by:

  • an algebraic formula, possibly piecewise,

  • a graph,

  • an experimentally determined table of values (a data table that can be interpolated by a curve if appropriate).

• A function need not be given by a single formula; e.g., a velocity function for a car may be piecewise defined over different time intervals.

• In many contexts, letters in the problem denote different roles:

  • The independent variable (the input) is typically (x) or time (t).

  • The dependent variable (the output) is typically (y) or velocity (v).

• When translating from words to mathematics in story problems, choose letters to represent variables carefully; there are traditions (e.g., time as (t), area as (A), volume as (V)).

Independent vs. dependent variables

• Independent variable: the variable you control or choose, usually denoted by (x) (or (t) for time).

• Dependent variable: the quantity that depends on the independent variable, usually denoted by (y) (or (v) for velocity).

• If a problem uses letters other than x and y, it’s usually to reflect the physical meaning (e.g., radius r, height h, time t).

Important example: Open-top box (Example 1.3.1)

• Setup: start with an (L \times W) rectangle of cardboard; cut out squares of side length (x) from each corner; fold up the sides to form a box with no top.

• Base dimensions after cutting: ((L - 2x) \times (W - 2x)).

• Height of the box: (x).

• Volume as a function of (x):

  • V(x) = x\,(L - 2x)(W - 2x).

• Domain of the volume function (physical constraints):

  • (x > 0) (you must cut out a positive square), and

  • (L - 2x > 0) and (W - 2x > 0) (the base dimensions must be positive),

  • so 0 < x < \min\left{\frac{L}{2}, \frac{W}{2}\right}.

• In interval notation the domain is ((0, \min{L/2, W/2})). The endpoints would make the base degenerate (zero area) and are not included for a real box.

Example: Circle centered at the origin (Example 1.3.2)

• Circle equation: x^2 + y^2 = r^2.

• Solving for (y) gives two branches: y = \pm\sqrt{r^2 - x^2}.

• To obtain a function, pick one sign (one branch).

• Upper semicircle as a function: y = f(x) = \sqrt{r^2 - x^2}.

• Domain of the upper semicircle function: must satisfy (r^2 - x^2 \ge 0), i.e. (-r \le x \le r). In interval notation: ([-r, r]).

• If we instead take the lower semicircle with the negative sign, domain is the same, but the graph is the lower half.

Domain considerations for square-root expressions (Example 1.3.3)

• Question: find the domain of a function involving a square root, e.g. something like a formula that includes (\sqrt{4x - x^2}) in the denominator (context may introduce division by this square root).

• Requirement: the expression under the square root must be positive (to avoid taking the square root of a negative number) and, if the square root occurs in a denominator, the square root must be nonzero.

• Determine when (4x - x^2 > 0):

  • Factor: (4x - x^2 = x(4 - x)), so the product is positive when either both factors are positive or both are negative.

  • Since (x) and (4 - x) have opposite signs outside ((0, 4)), the condition becomes (0 < x < 4).

• Alternative method (complete the square): (4x - x^2 = 4 - (x - 2)^2); positivity requires ((x - 2)^2 < 4) which also yields (0 < x < 4).

• Conclusion: domain in interval notation is ((0, 4)) (endpoints excluded if the expression would be in a denominator or otherwise undefined at the ends).

Domain in story problems and variable choices

• In story problems, the domain may be restricted beyond the algebraic domain due to practicality or physical meaning.

  • Example: area of a square with side length (s): (A = s^2). Although algebraically (s) could be any real number, in the context of area we require (s > 0) (a nonpositive side length is not meaningful for a square).

  • Example: if (V) is the volume of a sphere of radius (r): V = \frac{4}{3}\pi r^3, the domain is typically (r \ge 0) in physical problems (negative radii are not meaningful).

• In general, the independent variable is the quantity you control (often time (t) in motion problems); the dependent variable is the quantity that results from the change in the independent variable (e.g., velocity (v(t))).

• When choosing letters, it’s important to map the real-world meaning to mathematical symbols (e.g., (t) for time, (r) for radius, (h) for height).

Piecewise and nonformula representations

• Not all functions are given by a single algebraic formula. For example, a velocity function might be defined piecewise over time:

  • accelerate from 0 to 20 m/s in the first 10 s, then hold at 20 m/s for 15 s, then brake from 20 m/s to 0 in 5 s.

  • This yields a function that is defined by different expressions on different time intervals.

• Functions can also be given by a table of values (experimental data) or by a description that is not a standard formula.

Two key methods to find the domain of a function (illustrative)

• Method 1: Factor or analyze the inequality to ensure positivity/definedness (examples with square roots or denominators).

  • Example: For (4x - x^2 > 0) derive (0 < x < 4).

• Method 2: Complete the square to transform a condition into a standard inequality like ( (x - a)^2 < b^2 ) and solve for x accordingly.

Composition of functions (brief mention)

• The composition of two functions is defined as

  • ( (g\circ f)(x) = g(f(x)) )

  • The domain of the composition is constrained by where f(x) is defined and where g is defined on its input.

• Similarly for ( (f\circ g)(x) = f(g(x)) ).

Selected exercises and examples (Overview from the exercises in the text)

• Ex 1.3.1: Domain of the open-top box volume function, with a rectangle of dimensions (L) by (W):

  • Volume: V(x) = x\,(L - 2x)(W - 2x).

  • Domain: 0 < x < \min\left(\frac{L}{2}, \frac{W}{2}\right).

• Ex 1.3.2: Circle of radius (r) centered at the origin; for the upper semicircle, domain is ([-r, r]).

  • Function: y = f(x) = \sqrt{r^2 - x^2},\quad -r \le x \le r.

• Ex 1.3.3: Domain for a function involving a square root, likely with a denominator; the domain comes from requiring (4x - x^2 > 0) (and possibly excluding endpoints if they cause division by zero). Result: 0 < x < 4.

• Ex 1.3.4: Velocity function over time with piecewise segments; domain is the total time interval of motion (e.g., from time 0 to 30 s, if the acceleration to 20 m/s takes 10 s, then 15 s at 20 m/s, then 5 s braking).

• Ex 1.3.11–1.3.16: Additional problems including:

  • Domain of various functions, composition domains, and threshold values in story problems.

  • A farmer’s fence along a river: with 500 ft of fencing and a rectangular pen on three sides, x is the side perpendicular to the river; area A(x) = x(500 - 2x); domain: 0 \le x \le 250.

  • Cylinder can with fixed material in the side, top, and bottom (surface area fixed): find V as a function of radius r and the domain.

  • Cylinder optimization with fixed volume: minimize surface area for a can of volume 1000 cm^3; derive S(r) and find the radius that minimizes S (domain is (r > 0)).

  • Example expressions include: base and height relationships, sphere/box dimensions, and composition domains.

Quick recap: key concepts to remember

• A function assigns exactly one output for each input in its domain.

• The domain is the set of all valid inputs; it can be all real numbers or a restricted interval or union of intervals depending on the formula and context.

• Endpoints in domain notation are included with [ ] and excluded with ( ).

• The square-root and division by zero are common sources of domain restrictions; a square root requires a nonnegative radicand, and division by zero is undefined.

• In real-world problems, domain restrictions often reflect physical or practical constraints (e.g., nonnegative lengths, nonnegative time).

• The independent variable is typically the input (often x or t); the dependent variable is the output (often y or v).

• Functions can be defined in multiple ways: algebraic formulas, graphs, tables, or descriptive rules; piecewise definitions are common.

• The composition of functions has its own domain considerations and is defined by how the inner function feeds into the outer function.

Notation and conventions (summary)

• Interval notation uses: [a, b], [a, b), (a, b], (a, b) with brackets meaning inclusion and parentheses meaning exclusion.

• Common domains encountered:

  • [0, \infty) for (\sqrt{x}) or similar nonnegative-domain functions.

  • (-\infty, \infty) for functions defined on all real numbers like (f(x) = x^2).

  • (-r, r) or [-r, r] for circle-related expressions depending on whether endpoints are included.

• Mathematical expressions are written in LaTeX between double dollar signs for clarity, e.g. V(x) = x\,(L - 2x)(W - 2x) or 0 < x < \min\left(\frac{L}{2}, \frac{W}{2}\right).