Types of Functions
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Introduction to Function Families
In precalculus, we study different "families" of functions. Each family has its own characteristic shape, properties, and behaviors. Understanding these families helps you recognize patterns and solve problems efficiently.
1. Linear Functions
Linear Function
f(x) = mx + b
- m = slope (rate of change)
- b = y-intercept
- Graph: Straight line
- Domain: All real numbers
- Range: All real numbers
2. Quadratic Functions
Quadratic Function
f(x) = ax² + bx + c (standard form)
f(x) = a(x - h)² + k (vertex form)
- Vertex: (h, k) - the turning point
- Graph: Parabola (U-shaped)
- Opens up if a > 0; Opens down if a < 0
- Domain: All real numbers
- Range: [k, ∞) if a > 0; (-∞, k] if a < 0
3. Polynomial Functions
Polynomial Function
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
- Degree n determines the maximum number of zeros (n) and turning points (n-1)
- Graph: Smooth, continuous curve
- End behavior depends on degree and leading coefficient
- Domain: All real numbers
4. Rational Functions
Rational Function
f(x) = P(x)/Q(x), where P and Q are polynomials
- Vertical asymptotes: Where Q(x) = 0 (and P(x) ≠ 0)
- Horizontal asymptote: Depends on degrees of P and Q
- Holes: Where both P(x) = 0 and Q(x) = 0
- Domain: All real numbers except where Q(x) = 0
5. Radical (Root) Functions
Square Root Function
f(x) = √x or f(x) = a√(x - h) + k
- Starting point: (h, k)
- Domain: [h, ∞) for √(x-h)
- Range: [k, ∞) if a > 0; (-∞, k] if a < 0
Cube Root Function
f(x) = ∛x
- Domain: All real numbers
- Range: All real numbers
6. Exponential Functions
Exponential Function
f(x) = abˣ or f(x) = a · bˣ⁻ʰ + k
- Base b > 1: Exponential growth
- Base 0 < b < 1: Exponential decay
- Horizontal asymptote: y = k
- Domain: All real numbers
- Range: (k, ∞) if a > 0; (-∞, k) if a < 0
7. Logarithmic Functions
Logarithmic Function
f(x) = logᵦ(x) or f(x) = a · logᵦ(x - h) + k
- Inverse of: Exponential function with same base
- Vertical asymptote: x = h
- Domain: (h, ∞)
- Range: All real numbers
8. Absolute Value Functions
Absolute Value Function
f(x) = |x| or f(x) = a|x - h| + k
- Vertex: (h, k)
- Graph: V-shaped
- Opens up if a > 0; Opens down if a < 0
- Domain: All real numbers
- Range: [k, ∞) if a > 0; (-∞, k] if a < 0
9. Piecewise Functions
Piecewise Function
A function defined by different formulas on different intervals:
f(x) = { formula₁, if condition₁
{ formula₂, if condition₂
{ ...
Each "piece" may be from a different function family.
Identifying Function Types from Graphs
| If the graph... | It's likely a... |
|---|---|
| Is a straight line | Linear function |
| Is U-shaped (parabola) | Quadratic function |
| Has multiple turns, smooth | Polynomial function |
| Has vertical asymptote(s) | Rational or logarithmic |
| Has horizontal asymptote, curves | Exponential or rational |
| Is V-shaped | Absolute value function |
| Starts at a point, curves to the right | Square root function |
Examples
Example 1: Identifying Function Types
Problem: Identify the function type and key features of f(x) = 3(2)ˣ - 4
Solution:
This is an exponential function with:
- Base b = 2 > 1, so it represents exponential growth
- Vertical stretch by factor of 3
- Vertical shift down 4 units
- Horizontal asymptote: y = -4
- Domain: (-∞, ∞)
- Range: (-4, ∞)
- y-intercept: f(0) = 3(1) - 4 = -1
Example 2: Identifying from an Equation
Problem: Classify each function:
a) f(x) = (x² - 4)/(x + 2)
b) g(x) = 2√(x + 3) - 1
Solution:
a) This looks like a rational function, but let's simplify:
f(x) = (x-2)(x+2)/(x+2) = x - 2 (when x ≠ -2)
This is actually a linear function with a hole at x = -2.
b) This is a square root function with:
- Horizontal shift: left 3
- Vertical stretch by 2
- Vertical shift: down 1
- Starting point: (-3, -1)
- Domain: [-3, ∞)
- Range: [-1, ∞)
Example 3: Domain and Range
Problem: Find the domain and range of f(x) = log₂(x - 1) + 3
Solution:
This is a logarithmic function.
Domain: The argument must be positive:
x - 1 > 0 → x > 1
Domain: (1, ∞)
Range: Logarithmic functions have range of all real numbers.
Range: (-∞, ∞)
Vertical asymptote: x = 1
Example 4: Piecewise Function
Problem: Graph and find the domain and range of:
f(x) = { x + 2, if x < 0
{ x², if x ≥ 0
Solution:
For x < 0: Linear function (line with slope 1, y-intercept 2)
For x ≥ 0: Quadratic function (parabola starting at origin)
Domain: All real numbers (-∞, ∞)
Range: The linear piece approaches 2 as x → 0⁻. The quadratic piece starts at 0. So range is [0, ∞)
Note: There's a jump discontinuity at x = 0.
Example 5: Comparing Function Growth
Problem: Which grows faster as x → ∞: f(x) = x³ or g(x) = 2ˣ?
Solution:
Let's compare values:
| x | x³ | 2ˣ |
|---|---|---|
| 5 | 125 | 32 |
| 10 | 1,000 | 1,024 |
| 15 | 3,375 | 32,768 |
| 20 | 8,000 | 1,048,576 |
Exponential functions always eventually grow faster than polynomial functions.
This is because exponential growth compounds multiplicatively while polynomial growth adds powers.
Practice
Identify the function type and find domain and range for each.
1. f(x) = -2x + 5
2. g(x) = (x + 3)² - 4
3. h(x) = 1/(x - 2)
4. f(x) = √(4 - x)
5. g(x) = 5(0.5)ˣ
6. h(x) = ln(x + 1)
7. f(x) = |x - 4| + 2
8. g(x) = ∛(x - 1)
9. Classify: f(x) = (x³ - 8)/(x - 2)
10. Write a piecewise function that equals x² for x < 2 and equals 2x for x ≥ 2.
Click to reveal answers
- Linear; Domain: (-∞, ∞); Range: (-∞, ∞)
- Quadratic; Domain: (-∞, ∞); Range: [-4, ∞)
- Rational; Domain: (-∞, 2) ∪ (2, ∞); Range: (-∞, 0) ∪ (0, ∞)
- Square root; Domain: (-∞, 4]; Range: [0, ∞)
- Exponential (decay); Domain: (-∞, ∞); Range: (0, ∞)
- Logarithmic; Domain: (-1, ∞); Range: (-∞, ∞)
- Absolute value; Domain: (-∞, ∞); Range: [2, ∞)
- Cube root; Domain: (-∞, ∞); Range: (-∞, ∞)
- Simplifies to x² + 2x + 4 (with hole at x = 2) - Quadratic
- f(x) = { x², if x < 2; 2x, if x ≥ 2 }
Check Your Understanding
1. How do exponential and logarithmic functions relate to each other?
Show answer
They are inverse functions of each other. If f(x) = bˣ, then f⁻¹(x) = logᵦ(x). This means: if y = bˣ, then x = logᵦ(y). Graphically, they are reflections of each other over the line y = x.
2. Why does a square root function have a restricted domain but a cube root function doesn't?
Show answer
Square roots of negative numbers are not real (they're imaginary), so √x requires x ≥ 0. However, cube roots of negative numbers are real (for example, ∛(-8) = -2), so ∛x is defined for all real x.
3. What creates a vertical asymptote in a rational function?
Show answer
A vertical asymptote occurs where the denominator equals zero and the numerator doesn't equal zero at the same point. If both numerator and denominator equal zero at a point, that creates a hole instead of an asymptote.
4. Arrange in order from slowest to fastest growth as x → ∞: logarithmic, exponential, polynomial, linear.
Show answer
From slowest to fastest: logarithmic < linear < polynomial < exponential. Logarithmic growth is incredibly slow, exponential growth is incredibly fast, and polynomial growth depends on degree but is always eventually surpassed by exponential.
🚀 Next Steps
- Review any concepts that felt challenging
- Move on to the next lesson when ready
- Return to practice problems periodically for review