Functions

1. What is a Function?

A function is like a number machine. You put a number in (the input), the machine does something to it, and you get a number out (the output).

We write functions using notation like f(x) or g(x), where:

• f or g is the name of the function
• x is the input variable
• f(x) represents the output

Example: If f(x) = 2x + 3, this function takes any number x, multiplies it by 2, then adds 3.

• f(1) = 2(1) + 3 = 5
• f(4) = 2(4) + 3 = 11
• f(k) = 2k + 3

2. Domain and Range

Domain

The domain is the set of all possible input values (x-values) that you can put into a function. It's all the values of x for which the function "works" or is defined.

We often write x ∈ ℝ, which means "x belongs to the real numbers" (x can be any number).

Finding the domain:

For most functions, you can use any value of x. However, there are two important restrictions:

Rule 1: Square roots

• You cannot take the square root of a negative number
• So if f(x) = √(expression), the expression inside must be ≥ 0

Example: Find the domain of f(x) = √(2x + 3)

• We need: 2x + 3 ≥ 0
• 2x ≥ -3
• x ≥ -3/2
• Domain: x ≥ -3/2, x ∈ ℝ

Rule 2: Fractions

• You cannot divide by zero
• So if f(x) = (numerator)/(denominator), the denominator cannot equal zero

Example: Find the domain of f(x) = (2x + 5)/(3x - 1)

• We need: 3x - 1 ≠ 0
• 3x ≠ 1
• x ≠ 1/3
• Domain: x ≠ 1/3, x ∈ ℝ

Range

The range is the set of all possible output values (y-values) that the function can produce. It's all the values that f(x) can be.

Golden Rule: Never determine the range without looking at the graph of the function.

Think of it like a milkshake blender:

• The blender (function) processes ingredients (inputs)
• You cannot tell the flavor (range) just by looking at the machine
• You need to see what comes out!

Example: Find the range of f(x) = 2(x - 3)² + 5 for x ∈ ℝ

This is a quadratic function (a parabola). When we sketch it:

• The turning point (minimum point) is at (3, 5)
• The parabola opens upwards
• The lowest y-value is 5
• The function can produce any value greater than or equal to 5

Range: f(x) ≥ 5 or y ≥ 5

Range with Restricted Domains

When the domain is restricted (limited to certain x-values), the range changes accordingly. You must:

1. Find the y-values at the boundary points of the domain
2. Find the turning point (if it's within the domain)
3. Determine the minimum and maximum y-values

Example 1: f(x) = 2(x - 3)² + 5 for x ≥ 1

• At x = 1: f(1) = 2(1 - 3)² + 5 = 2(4) + 5 = 13
• Turning point at x = 3 (this is within the domain x ≥ 1)
• At turning point: f(3) = 5
• Since the parabola opens upward, the minimum is at the turning point
• Range: f(x) ≥ 5

Example 2: f(x) = 2(x - 3)² + 5 for x < 1

• At x = 1: f(1) = 13 (but x = 1 is NOT included, so we use >)
• Turning point at x = 3 is NOT in this domain
• As x decreases from 1, f(x) increases
• Range: f(x) > 13

Example 3: f(x) = 2(x - 3)² + 5 for 4 < x ≤ 5

• At x = 4: f(4) = 2(4 - 3)² + 5 = 7 (not included, use >)
• At x = 5: f(5) = 2(5 - 3)² + 5 = 13 (included, use ≤)
• Turning point at x = 3 is NOT in this domain
• Range: 7 < f(x) ≤ 13

Important examples from syllabus:

Example A: f: x → 1/x for x ≥ 1

• At x = 1: f(1) = 1
• As x increases, 1/x decreases towards 0 (but never reaches it)
• Range: 0 < f(x) ≤ 1

Example B: g: x → x² + 1 for x ∈ ℝ

• This is a parabola with turning point at (0, 1)
• Minimum value is g(0) = 1
• The parabola opens upward
• Range: g(x) ≥ 1