Types of functions: every family has its own domain

The problem

Type x = 90° into tan on your calculator and it says “Math ERROR”. Type −5 into log and you get the same complaint. Neither your calculator nor you did anything wrong: some functions simply refuse certain inputs.

Functions come in families, and each family has its own rules about which inputs it accepts (the domain) and which outputs it can give (the range). Knowing the family tells you the domain before you even calculate.

Algebraic functions are built from algebraic expressions: polynomial, linear, identity, constant and rational functions. Trigonometric, inverse trigonometric, exponential and logarithmic functions are the other families in this section.

Key terms

Polynomial and rational

A polynomial has non-negative integer powers of x; if aₙ ≠ 0 its degree is n and aₙ is the leading coefficient. A rational function is P(x)/Q(x); its domain is every x with Q(x) ≠ 0. Linear, identity and constant functions are special polynomials.

Trigonometric

sin x and cos x accept every real x and give values from −1 to 1. tan x and sec x reject x = (2n + 1)π/2; cot x and csc x reject x = nπ. sec x and csc x never lie strictly between −1 and 1.

Inverse trigonometric

y = sin⁻¹x means x = sin y with −π/2 ≤ y ≤ π/2 and −1 ≤ x ≤ 1. y = cos⁻¹x uses 0 ≤ y ≤ π. y = tan⁻¹x uses −π/2 < y < π/2 and accepts every real x.

Exponential and logarithmic

In an exponential function the variable is in the power: eˣ, eᵃˣ, 2ˣ = e^(x ln 2). If x = aʸ then y = logₐx (a > 0, a ≠ 1); base 10 gives lg x and base e gives ln x.

Short questions with model answers

1. Q1. State the degree and leading coefficient of P(x) = 2x⁴ − 3x³ + 2x − 1, and say why it is a polynomial function.

   • powers: 4, 3, 1, 0
   • highest power 4, coefficient 2 ≠ 0

   degree 4 · leading coefficient 2

2. Q2. Find the domain of the rational function R(x) = (2x + 1)/(x² − 5x + 6).

   • x² − 5x + 6 = (x − 2)(x − 3)
   • Q(x) = 0 ⇔ x = 2 or x = 3

   Domain R = ℝ − {2, 3}

3. Q3. Write 2ˣ as a power of e, and check your answer at x = 3.

   • 2 = e^(ln 2) ⇒ 2ˣ = e^(x ln 2)
   • x = 3: e^(3 × 0.6931) = e^2.079 = 8.00 = 2³

   2ˣ = e^(x ln 2)

Common mistakes

• ✗ Writing the range of sec x as ℝ, as it is printed in the book.

  ✓ sec x = 1/cos x, and |cos x| ≤ 1, so |sec x| ≥ 1. The range is y ≥ 1 or y ≤ −1, exactly like csc x.

• ✗ Calling x² + 1/x a polynomial because it has powers of x.

  ✓ 1/x = x⁻¹ has a negative exponent. A polynomial needs non-negative integer exponents only.

• ✗ Reading sin⁻¹x as 1/sin x.

  ✓ sin⁻¹x is the angle whose sine is x. The reciprocal 1/sin x is csc x, a different function.

MCQs

1. 1. The function f(x) = 3x + 4 is:

   1. (a) constant
   2. (b) identity
   3. (c) linear
   4. (d) rational
   Show answer

   (c) A polynomial of degree 1, f(x) = ax + b with a ≠ 0, is linear.

2. 2. Which inputs does tan x refuse?

   1. (a) x = nπ
   2. (b) x = (2n + 1)π/2
   3. (c) x < 0
   4. (d) none
   Show answer

   (b) tan x = sin x/cos x, and cos x = 0 at x = (2n + 1)π/2.

3. 3. The range of y = sin⁻¹x is:

   1. (a) [0, π]
   2. (b) [−π/2, π/2]
   3. (c) (−π/2, π/2)
   4. (d) [−1, 1]
   Show answer

   (b) −π/2 ≤ y ≤ π/2. [0, π] belongs to cos⁻¹x, and [−1, 1] is the domain, not the range.

4. 4. Which of these is a polynomial function?

   1. (a) x² + 1/x
   2. (b) √x + 1
   3. (c) 5x³ − x + 7
   4. (d) 2ˣ
   Show answer

   (c) 5x³ − x + 7 has only non-negative integer powers. 1/x and √x have non-integer or negative powers; 2ˣ is exponential.

5. 5. lg x means the logarithm of x to the base:

   1. (a) e
   2. (b) 2
   3. (c) 10
   4. (d) x
   Show answer

   (c) Base 10, the common logarithm. Base e gives the natural logarithm, ln x.

Quick revision

• Algebraic functions: polynomial (degree, leading coefficient), linear, identity, constant, rational (domain: Q(x) ≠ 0).
• tan and sec skip x = (2n + 1)π/2; cot and csc skip x = nπ; sec and csc have range |y| ≥ 1.
• sin⁻¹x, cos⁻¹x, tan⁻¹x have restricted ranges; aˣ and logₐx undo each other (lg: base 10, ln: base e).