FSc Part 1 Mathematics Chapter 2 Complete Guide Book — Functions and Graphs

The word function was used by the German mathematician Leibniz (1646–1716) to describe how one quantity depends on another. The book opens with two everyday examples. The area A of a square depends on its side x by the formula A = x², so we say A is a function of x: give me the side, and the area is fixed — one side can never give two different areas. Likewise the volume V of a sphere depends on its radius r by V = (4/3)πr³, so V is a function of r.

That “one and only one” idea is the whole definition. A function is a rule of correspondence relating two sets in such a way that each element in the first set corresponds to one and only one element in the second set. If one input tried to point at two different outputs, the rule would not be a function at all.

2.1.1 — Definition (Function, Domain, Codomain, Range). A function f from a set X to a set Y is a rule that assigns to each element x in X a unique element y in Y. The set X is called the domain of f — the set of all allowed inputs. The set of corresponding elements y in Y is called the range of f — the outputs that actually occur. The whole set Y is the codomain — the set in which the outputs live. Unless stated otherwise, the book assumes X and Y consist of real numbers.

📌 Book note (p. 23): the codomain is the set of all possible outputs, but the range is the actual set of outputs produced under the given domain — so the range is always a subset of the codomain. For f(x) = x² with domain all reals, the codomain may be all reals, but the range is only the non-negative reals, because a square is never negative.

If a variable y depends on x so that each value of x determines exactly one value of y, we say “y is a function of x”. The Swiss mathematician Euler (1707–1783) invented the short way of writing this: y = f(x), read as “y is equal to f of x”. Think of f as a computing machine: it takes an input x, operates on it in some way, and produces exactly one output f(x). That output is called the value of f at x, or the image of x under f. Here x is the independent variable (you choose it) and y is the dependent variable (it depends on x). From now on the book considers only functions whose variables are real numbers — a real-valued function of real numbers.

Example 1 (book, p. 23). Given f(x) = x³ − 2x² + 4x − 1, find f(0), f(1), f(−2), f(1 + x) and f(1/x). The method never changes: wherever you see x, put the new value in brackets and simplify. So f(0) = −1; f(1) = 1 − 2 + 4 − 1 = 2; f(−2) = −8 − 8 − 8 − 1 = −25; f(1 + x) expands to x³ + x² + 3x + 2; and f(1/x) = 1/x³ − 2/x² + 4/x − 1 (x ≠ 0). The f(−2) line is where sign slips happen — write (−2)³ = −8 and −2(−2)² = −8 separately before adding.

Example 2. For f(x) = x², every real x gives a non-negative output, so Domain f = set of all real numbers; Range f = set of all non-negative real numbers. That is the codomain-versus-range distinction in action.

Example 3 (p. 24). For f(x) = x/(x² − 4), the denominator is zero at x = 2 and x = −2, so those two inputs are forbidden: Domain = R − {−2, 2}. To find the range, set y = x/(x² − 4) and solve for x: x²y − x − 4y = 0, so x = [1 ± √(1 + 16y²)]/(2y) for y ≠ 0, which is defined for every y ≠ 0; and y = 0 occurs at x = 0 (f(0) = 0). Hence Range = all real numbers, (−∞, ∞).

Example 4. For f(x) = √(x² − 9), a square root of a negative number is not real, so we need x² − 9 ≥ 0. Testing the three regions cut by x = −3 and x = 3 (x = −4 works, x = 0 fails, x = 4 works) gives Domain = (−∞, −3] ∪ [3, ∞). The smallest value of x² − 9 is 0 (at x = ±3), and it grows without bound, so Range = [0, ∞).

📌 Book note (Remember box, p. 24): in an open interval (a, b) the endpoints are not included; in a closed interval [a, b] they are included. That is why the domain above uses square brackets at −3 and 3 — those points are allowed.

2.1.3 — Vertical Line Test. A graph represents a function if and only if no vertical line intersects the graph more than once. If any vertical line cuts the graph twice or more, one x would have two y-values, so it is not a function. The book’s figure (p. 25) shows this cleanly: a wavy curve and a straight line pass the test (functions), while a circle and a sideways S-curve fail it (not functions) because a vertical line meets them twice.

2.1.4 — Types of Function. (i) One-to-one (injective): different inputs produce different outputs — if f(x₁) = f(x₂) then x₁ = x₂. For example f(x) = 5x + 7 is one-to-one, since 5x₁ + 7 = 5x₂ + 7 forces x₁ = x₂. (ii) Onto (surjective): every element of the codomain Y has at least one pre-image in the domain — for every y in Y there is an x with f(x) = y. For example f(x) = 2x + 3 on R is onto: given any y, x = (y − 3)/2 works, and f((y − 3)/2) = y. (iii) Bijective: both one-to-one and onto at once.

Piecewise Function. A function defined by different expressions (“pieces”) over different intervals of its domain. The book’s example is f(x) = 2x + 1 for x < 0, and x² − 1 for x ≥ 0: left of zero the graph follows the line, from zero onwards it follows the parabola. When you evaluate a piecewise function, first ask which interval x falls in, then use only that piece.

Example 5 (p. 26). f(x) = x + 1 on R is bijective. One-to-one: f(x₁) = f(x₂) gives x₁ + 1 = x₂ + 1, so x₁ = x₂. Onto: for any real y, x = y − 1 gives f(y − 1) = y. Both hold, so it is bijective.

Example 6. f(x) = x² − 2 on R is neither one-to-one nor onto. Not one-to-one: f(2) = 2 = f(−2) with 2 ≠ −2. Not onto: no real x gives f(x) = −3 (that would need x² = −1), and indeed every value below −2 is missed — −2 is the smallest value, attained only at x = 0.

2.2 — Finding the Intersecting Point(s) Graphically. The point of intersection is where two or more graphs meet on the coordinate plane. That point matters because it represents the solution of the equations of the given functions: at the meeting point, both functions have the same x and the same y.

2.2.1 — Intersection of a Linear Function and Coordinate Axes. A linear function is one in which the highest power of the variable is one, and the coordinate axes are the x-axis and y-axis of the Cartesian system. The book’s method is deliberately simple: make a table of values, plot the points, draw the line, and read off where it crosses each axis.

Example 7 (p. 28). Find the intersection points of y = 2x + 6 with the axes, graphically. Table: x = −1 gives y = 4; x = 0 gives y = 6; x = 1 gives y = 8. Plot those points and the line crosses the x-axis at (−3, 0) and the y-axis at (0, 6). You can also check without the graph: set y = 0 for the x-intercept (0 = 2x + 6 gives x = −3) and x = 0 for the y-intercept (y = 6). The graph and the algebra must agree — if they do not, recheck the table.

The point of intersection of two linear functions is the point where their graphs cross each other — the one point where both functions share the same x and the same y. Graphically you plot both lines from a table of values and read the crossing point; algebraically you set the two expressions equal, because at the crossing point the y-values are equal.

Example 8 (p. 28). Find the intersection of y = 3x + 2 and y = −x + 6, graphically. Tables: for y = 3x + 2, x = −1, 0, 1 give y = −1, 2, 5; for y = −x + 6 the same x-values give y = 7, 6, 5. Both lines pass through (1, 5), so that is the intersection point. Check by algebra: 3x + 2 = −x + 6 gives 4x = 4, x = 1, and then y = 5. Again, graph and algebra agree — and in Exercise 2.2 you will use exactly this double check for every pair.

A line and a parabola can meet in two points, one point, or not at all. Two solutions mean two intersection points. A single solution means only one intersection point — the line may be tangent to the parabola, just touching it. No solution means the line and the parabola do not intersect. The book’s figure (p. 29) shows all three cases side by side, and that picture is worth remembering before you solve anything: the number of solutions of the combined equation is the number of meeting points.

Example 9 (p. 29). Solve y = −x + 3 and y = x² − 6x + 3 graphically. First the line: its x-intercept is (3, 0) and its y-intercept is (0, 3). Then the parabola: at x = 0, y = 3, so (0, 3) is its y-intercept; setting y = 0 gives x² − 6x + 3 = 0, so x = 3 ± √6 ≈ 0.6 and 5.4, giving x-intercepts (0.6, 0) and (5.4, 0). Its vertex (h, k) comes from h = −b/(2a) = 3 and k = 9 − 18 + 3 = −6, so the vertex is (3, −6). Plotting both gives the intersection points (0, 3) and (5, −2) — check them: for (5, −2), the line gives −5 + 3 = −2 and the parabola gives 25 − 30 + 3 = −2. Both agree.

Example 10 (p. 30). Graph y = 2√x + 1. The domain is x ≥ 0, because the square root of a negative number is not a real number, and the range is y ≥ 1, since 2√x is never negative and then 1 is added. The book’s table runs from x = 0 (y = 1) through x = 1 (y = 3), x = 4 (y = 5) and x = 9 (y = 7) up to x = 10 (y ≈ 7.3). Plot those points and you get a curve that starts at (0, 1), rises steeply at first and then flattens — the classic square-root shape. Two things to notice: the curve never goes left of the y-axis (that is the domain, x ≥ 0, drawn on the graph), and equal steps in x give smaller and smaller steps in y, because √x grows more slowly as x grows.

Example 11 (p. 30). Graph y = ∛(x − 1). The big difference from the square root is that a cube root is defined for negative numbers — (−2)³ = −8, so ∛(−8) = −2 — so this graph runs on both sides of the y-axis. The book’s table: x = −5 gives y ≈ −1.8, x = −1 gives y ≈ −1.3, x = 0 gives y = −1, x = 1 gives y = 0, x = 2 gives y = 1, and x = 5 gives y ≈ 1.6. Joining the points gives the gentle S-shape of a cube-root curve, crossing the x-axis at (1, 0). If you compare it with the square-root curve of the previous section, the contrast is the lesson: square roots stop at zero and go one way only; cube roots pass straight through and keep going both ways.

When something increases in quantity or size over time it is called growth — money in a bank account earning interest, or a population of rabbits increasing over months. When something decreases over time it is called decay — a radioactive substance losing its strength over years, or a cup of hot coffee cooling down. Both are modelled by functions of time, and the book works two of them out in full.

Example 12 (p. 31). A stock follows P(t) = P₀e^(rt), with P₀ the initial price, r the growth rate per year and t the time in years. A stock worth Rs. 5,000 growing at 5% per year: (i) after 10 years, P(10) = 5000e^(0.05×10) = 5000e^0.5 ≈ 5000 × 1.6487 = Rs. 8,244. (ii) Doubling: set P(t) = 2P₀, so 2 = e^(rt); taking natural logs, ln 2 = rt, so t = (ln 2)/r = 0.6931/0.05 ≈ 13.86 years, about 14 years.

Example 13 (p. 32). Pollutant concentration in a lake decays as C(t) = 100/√(t + 1) ppm, t in days. (i) After 4 days: C(4) = 100/√5 ≈ 44.72 ppm. (ii) Below 10 ppm: set 10 = 100/√(t + 1), so √(t + 1) = 10, t + 1 = 100, t = 99 days — strictly, the concentration equals 10 at 99 days and drops below 10 just after that.