Teaching Notes · Math for Social Sciences

Functions, Graphs, and Transformations

The first set of notes in this course defined a function as a rule assigning exactly one output to each input, and introduced domain, range, and the vertical line test. That vocabulary is enough to recognize a function, but not yet enough to work with one comfortably. This set of notes fills in the rest: a few named shapes of function worth being able to spot on sight, what it means to combine or chain two functions together, how to undo a function with its inverse, and how a graph's symmetry and its response to a shift or a flip can be read straight off its equation.

A few functions worth naming

A constant function, h(x) = c, hands back the same value no matter what goes in: for h(x) = 2, h(10) = 2, h(−387) = 2, and h(x + 3) = 2 as well, so its range is the single value {2}. A fixed monthly rent of 1000, R(x) = 1000 for any month x, is a constant function in exactly this sense: the output never moves. A rational function is simply a ratio of two polynomials, such as f(x) = (x² − 6x)/(x + 5), or even g(x) = 2x + 3, which counts as rational since it can be written as (2x + 3)/1. The absolute value function, f(x) = |x|, returns x itself when x is at least 0 and −x when x is negative; evaluated at a few points, f(x) = |2x − 1| gives f(0) = |−1| = 1, f(1) = |1| = 1, and f(0.5) = |0| = 0.

A case defined (piecewise) function states a different rule on different parts of its domain. Consider

f(x) = x for 0 ≤ x < 3
f(x) = x − 1 for 3 ≤ x ≤ 5
f(x) = 4 for 5 < x ≤ 7

Reading off a table of values, f(0) = 0, f(1) = 1, f(2) = 2, then f(3) = 2 (not 3, since x = 3 falls under the second rule), f(4) = 3, f(5) = 4, f(6) = 4, and f(7) = 4. The switch from the first rule to the second creates a visible jump: as x approaches 3 from the left the function climbs toward 3, but the value actually assigned at x = 3 is 2.

Combining and composing functions

Two functions can be added, subtracted, multiplied, or divided just like two numbers, simply by combining their output values: (f + g)(x) = f(x) + g(x), (f − g)(x) = f(x) − g(x), (fg)(x) = f(x)·g(x), and (f/g)(x) = f(x)/g(x) wherever g(x) is not 0. For f(x) = 3x − 1 and g(x) = x² + 3x, this gives (f + g)(x) = x² + 6x − 1, (fg)(x) = (3x − 1)(x² + 3x) = 3x³ + 8x² − 3x, and (f/g)(x) = (3x − 1)/(x² + 3x), undefined at x = 0 and x = −3, exactly where g(x) = 0.

Composition is a different way of combining two functions: instead of combining their outputs, the output of one becomes the input of the other. For g: X → Y and f: Y → Z, the composite (f ∘ g)(x) = f(g(x)) first applies g, then applies f to whatever g produced. With F(p) = p² + 4p − 3, G(p) = 2p + 1, and H(p) = |p|, composing F with G means substituting G(p) everywhere F has a p:

F(G(p)) = (2p + 1)² + 4(2p + 1) − 3 = (4p² + 4p + 1) + (8p + 4) − 3 = 4p² + 12p + 2

A composition with numbers plugged in first collapses even faster. Since F(1) = 1² + 4(1) − 3 = 2, then G(F(1)) = G(2) = 2(2) + 1 = 5, without ever needing a general formula for G(F(p)).

Inverse functions

A function is one to one when it never sends two different inputs to the same output: if f(a) = f(b), then a must equal b. Only a one to one function has an inverse, written f⁻¹(x), satisfying f⁻¹(f(x)) = x for every x in the domain of f and f(f⁻¹(y)) = y for every y in the range of f. Finding an inverse is mostly a matter of bookkeeping: write y = f(x), solve that equation for x in terms of y, and the resulting expression, with x and y swapped back, is f⁻¹(x).

Take f(x) = (x − 1)², restricted to x ≥ 1 (the restriction matters: without it, two different inputs on either side of x = 1 would produce the same output, and the function would not be one to one at all). Setting y = (x − 1)² and using x ≥ 1 to keep only the positive square root, x − 1 = √y, so x = √y + 1. Swapping the letters back gives f⁻¹(x) = √x + 1, a function that genuinely undoes f: feed it what f produced, and it returns the original input.

Symmetry

A graph's symmetry can be read directly from its equation, without plotting a single point. Three tests cover the usual cases: replacing y with −y and checking whether an equivalent equation results tests for symmetry about the x axis; replacing x with −x tests for symmetry about the y axis; replacing both at once tests for symmetry about the origin.

Apply all three to y = 1 − x⁴. Replacing y with −y gives −y = 1 − x⁴, which is not equivalent to the original equation, so there is no symmetry about the x axis. Replacing x with −x gives y = 1 − (−x)⁴ = 1 − x⁴, exactly the original equation, so the graph is symmetric about the y axis. Replacing both gives −y = 1 − x⁴ again, still not equivalent, so there is no symmetry about the origin either. The intercepts follow from the same equation: setting y = 0 gives 1 − x⁴ = 0, which factors as (1 − x)(1 + x)(1 + x²) = 0, so the x intercepts are (1, 0) and (−1, 0); setting x = 0 gives the single y intercept (0, 1).

Translations and reflections

Once the graph of a function is known, a short list of rules predicts what happens to it under a shift or a flip, without recomputing a single new point. Adding a positive constant c to f(x) shifts the whole graph up by c units; subtracting it shifts the graph down. Replacing x with x − c shifts the graph c units to the right, and replacing x with x + c shifts it c units to the left. Putting a minus sign in front of the whole function, −f(x), reflects the graph about the x axis, while replacing x with −x reflects it about the y axis.

Applied to y = (x − 1)³, the x has been replaced with x − 1, so this is the horizontal shift rule with c = 1: the graph of y = x³ moves one unit to the right. Checking a few points confirms it directly: the point (0, 0) on y = x³ moves to (1, 0), the point (1, 1) moves to (2, 1), the point (−1, −1) moves to (0, −1), and the point (−2, −8) moves to (−1, −8), every point on the familiar cubic curve carried one unit rightward without changing its shape at all.

Special functions, combinations, inverses, symmetry, and transformations are, in the end, four different ways of answering the same question: given a function already understood, what can be said about a new one built from it? That question comes up constantly in the rest of this course, every time a cost function is shifted by a fixed overhead, a demand function is reflected to read as a function of price instead of quantity, or a derivative is computed for a composition of two simpler functions.

Source: Ernest F. Haeussler Jr., Richard S. Paul, and Richard J. Wood, Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, 14th edition (Pearson Canada, 2019), chapter 2, covering special functions, combining and composing functions, inverse functions, symmetry, and translations and reflections of a graph.