Functions and Graphs

Subject: Mathematics
Topic: 2
Cambridge Code: 0580

---

Functions

Function - Relationship between input and output

Function Notation

$f(x) = 2x + 3$

Read as: "f of x equals 2x plus 3"

• Input: x
• Output: f(x)
• f(2) = 2(2) + 3 = 7

Domain and Range

Domain - Set of all possible input values

Range (Codomain) - Set of all possible output values

Example: $f(x) = \sqrt{x}$

• Domain: $x ≥ 0$ (cannot take square root of negative)
• Range: $f(x) ≥ 0$ (square roots are non-negative)

Types of Functions

Linear: $f(x) = mx + c$

• Straight line
• One solution usually

Quadratic: $f(x) = ax^2 + bx + c$

• Parabola
• Up to two solutions

Cubic: $f(x) = ax^3 + bx^2 + cx + d$

• S-shaped curve
• Up to three solutions

Rational: $f(x) = \frac{p(x)}{q(x)}$

• Discontinuous at zeros of denominator
• May have vertical asymptotes

Trigonometric: $f(x) = \sin x, \cos x, \tan x$

• Periodic functions
• Specific domains and ranges

Exponential: $f(x) = a^x$

• Continuous growth or decay
• Always positive

Logarithmic: $f(x) = \log_a x$

• Inverse of exponential
• Domain: x > 0

---

Inverse Functions

Inverse function - Undoes original function

Finding Inverse

Process:

1. Write $y = f(x)$
2. Swap x and y
3. Solve for y
4. Replace y with $f^{-1}(x)$

Example: $f(x) = 2x + 3$ $y = 2x + 3$ $x = 2y + 3$ $2y = x - 3$ $y = \frac{x - 3}{2}$ $f^{-1}(x) = \frac{x - 3}{2}$

Conditions for Inverse

Function must be:

• One-to-one (injective): Each output from exactly one input
• Onto (surjective): Every possible output is achieved

Graphical test: Horizontal line test

• Horizontal line intersects graph at most once

Property

$f(f^{-1}(x)) = x$ $f^{-1}(f(x)) = x$

---

Graph Transformations

Transformations - Changes to position, shape, or orientation

Translations (Shifts)

Horizontal shift:

• $f(x - h)$: Shift right by h units
• $f(x + h)$: Shift left by h units

Vertical shift:

• $f(x) + k$: Shift up by k units
• $f(x) - k$: Shift down by k units

Example: $y = (x - 2)^2 + 3$

• Parabola $y = x^2$ shifted 2 right, 3 up

Reflections

Reflection across x-axis:

• $y = -f(x)$: Flip upside down

Reflection across y-axis:

• $y = f(-x)$: Mirror image

Stretches and Compressions

Vertical stretch by factor a:

• $y = af(x)$: Stretches if |a| > 1
• $y = af(x)$: Compresses if 0 < |a| < 1

Horizontal stretch by factor a:

• $y = f(\frac{x}{a})$: Stretches if a > 1
• $y = f(\frac{x}{a})$: Compresses if 0 < a < 1

---

Curve Sketching

Key Features to Identify

Intercepts:

• x-intercepts (roots/zeros): Where $f(x) = 0$
• y-intercept: Value of $f(0)$

Asymptotes:

• Vertical: Lines function approaches (undefined)
• Horizontal: Lines function approaches as $x → ±∞$
• Oblique: Non-horizontal asymptotes

Turning points (extrema):

• Maximum: Highest point in region
• Minimum: Lowest point in region
• Stationary points: Where $f'(x) = 0$

End behavior:

• What happens as $x → +∞$?
• What happens as $x → -∞$?

Symmetry:

• Even function: $f(-x) = f(x)$ (symmetric about y-axis)
• Odd function: $f(-x) = -f(x)$ (symmetric about origin)

Sketching Process

1. Find domain and range
2. Find intercepts
3. Find asymptotes
4. Find turning points
5. Check symmetry
6. Determine end behavior
7. Sketch curve

---

Special Functions

Absolute Value Function

$f(x) = |x|$

• V-shaped graph
• Vertex at origin
• Domain: all real numbers
• Range: $y ≥ 0$

Piecewise Functions

$f(x) = \begin{cases} x^2 & \text{if } x < 0 \\ x & \text{if } x ≥ 0 \end{cases}$

• Different rules for different domains
• Graph may have corners or jumps
• Check continuity at boundaries

Modulus Function

$|f(x)|$

• Reflects negative part upward
• All outputs become non-negative

---

Composition of Functions

Composition - One function applied after another

$f∘g(x) = f(g(x))$

Example:

• $f(x) = 2x$
• $g(x) = x + 3$
• $f∘g(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6$

Note: $f∘g ≠ g∘f$ usually

---

Key Points

1. Function maps inputs to outputs
2. Domain: input values; Range: output values
3. One-to-one function has inverse
4. Inverse undoes original function
5. Translations shift graphs horizontally/vertically
6. Stretches change shape
7. Reflections flip graphs
8. Curve sketching requires identifying key features
9. Asymptotes show behavior at infinity
10. Composition applies functions in sequence

---

Practice Questions

1. Evaluate functions for given values
2. Find domain and range
3. Determine if function is one-to-one
4. Find inverse functions
5. Apply transformations
6. Sketch curves with transformations
7. Identify asymptotes
8. Analyze piecewise functions
9. Compose functions
10. Solve problems involving transformations

---

Revision Tips

• Practice finding domain and range
• Use coordinate grid for transformations
• Remember horizontal shift direction
• Learn asymptote finding techniques
• Practice curve sketching regularly
• Understand composition order
• Test inverse by composition
• Identify symmetry in functions