8 min readConcept sheetShort answer; part of a structured question

Functions: domain, range, inverse and graphs

A function gives exactly one output for each input. The domain is the set of allowed inputs; the range is the set of outputs it produces.

Notationy = f(x): f of x

Vertical line testNo vertical line cuts the graph twice

Domain trapsNo ÷ 0, no √ of a negative

InverseSwap x and y, then solve for y

Function or not?

A functionEach input has one output. Example: each product code at a Keells till gives one price; y = 2x + 1; y = x².

Not a functionSome input has two or more outputs. Example: x = y² (x = 4 gives y = 2 and y = −2).

Common functions and their graphs

Type	Form	Graph	Business example
Linear	y = mx + c	Straight line	Cost = fixed + variable
Quadratic	y = ax² + bx + c	Parabola (up if a > 0, down if a < 0)	Revenue with a falling price
Reciprocal	y = k/x	Hyperbola	Fixed cost per unit, FC/x
Exponential	y = abˣ	Growth or decay curve	Compound growth of a deposit
Inverse f⁻¹	Undoes f	Mirror image of f in the line y = x	Quantity from a price equation

Rules for domain and inverse

Fraction 1/(x − a)Domain: all real x except a

Square root √(x − a)Domain: x ≥ a; range: y ≥ 0

PolynomialDomain: all real numbers

Business modelDomain limited to sensible values, e.g. 0 ≤ x ≤ capacity

Inverse existsOnly if f is one-to-one (each output from one input)

Check an inversef(f⁻¹(x)) = x and f⁻¹(f(x)) = x

Finding an inverse function

Evaluate a function

Problem

f(x) = 3x² − 2x + 5. Find f(2), f(−1) and f(a + 1).

Working

f(2) = 3(4) − 2(2) + 5 = 12 − 4 + 5 = 13.
f(−1) = 3(1) − 2(−1) + 5 = 3 + 2 + 5 = 10.
f(a + 1) = 3(a + 1)² − 2(a + 1) + 5 = 3(a² + 2a + 1) − 2a − 2 + 5.
= 3a² + 6a + 3 − 2a − 2 + 5 = 3a² + 4a + 6.

Answer

f(2) = 13, f(−1) = 10, f(a + 1) = 3a² + 4a + 6.

Domain and range

Problem

State the domain (and range where asked): (a) f(x) = 1/(x − 4); (b) g(x) = √(x − 3), with its range; (c) a Matale workshop's cost C(x) = 5,000 + 200x for 0 ≤ x ≤ 100 units, with its range.

Working

(a) Division by zero is not allowed: x − 4 ≠ 0, so x ≠ 4. Domain: all real numbers except 4.
(b) The square root needs x − 3 ≥ 0, so x ≥ 3. A square root is never negative: range y ≥ 0.
(c) The domain is given: 0 ≤ x ≤ 100. Lowest cost C(0) = 5,000; highest C(100) = 5,000 + 20,000 = 25,000.

Answer

(a) x ≠ 4 (b) domain x ≥ 3, range y ≥ 0 (c) domain 0 ≤ x ≤ 100, range Rs. 5,000 to Rs. 25,000.

Find an inverse

Problem

(a) Find the inverse of f(x) = 2x + 6 and check it. (b) A demand function is p = 500 − 2q. Write q as a function of p.

Working

(a) y = 2x + 6. Swap: x = 2y + 6. Solve: 2y = x − 6, y = (x − 6)/2.
So f⁻¹(x) = (x − 6)/2. Check: f(5) = 16 and f⁻¹(16) = 10/2 = 5. ✓
(b) p = 500 − 2q, so 2q = 500 − p and q = 250 − 0.5p.
Check: p = 300 gives q = 100, and 500 − 2(100) = 300. ✓

Answer

(a) f⁻¹(x) = (x − 6)/2 (b) q = 250 − 0.5p

In Sri Lankan business

A CEB electricity bill or a payroll sheet is a function: each input (units used, hours worked) gives one amount due. The domain is the realistic inputs; units used can't be negative.

Exam angle

Short answer: “State the domain of f(x) = √(x − 2).” x − 2 ≥ 0, so x ≥ 2.

Short answer: “Find the inverse of f(x) = 5x + 2.” y = 5x + 2 → x = 5y + 2 → f⁻¹(x) = (x − 2)/5.

Structured question: “f(x) = 4x − 8: find f(3), the domain and range, and f⁻¹(x) with a check.” f(3) = 4; domain and range all reals; f⁻¹(x) = (x + 8)/4.

Remember it

Domain: ask 'what can I put in?' (no ÷ 0, no √ of a negative). Range: ask 'what can come out?' Inverse: swap, solve, check.