Mastering Rational, Exponential, and Logarithmic Functions

Quick Graph Identification

• Holes: Represented by open circles.
• Vertical Asymptotes: Represented by dashed vertical lines.
• Horizontal or Slant Asymptotes: Represented by dashed lines.
• Note: The graph approaches but never touches the asymptotes.

Understanding Domain

• Exclude values that make the denominator equal to zero.
• Even if a factor cancels, the value is still excluded from the domain.

Example: (x + 3) / (x(x + 3))
Domain: x ≠ -3, x ≠ 0

Final One-Pass Checklist

1. Factor and cancel.
2. Find holes.
3. Find vertical asymptotes.
4. Find x-intercepts.
5. Find the y-intercept.
6. Find horizontal or slant asymptotes.

Exponential and Logarithmic Functions

Exponential: f(x) = a · b^(x - h) + k (where b > 0, b ≠ 1)
Logarithmic: f(x) = a · log_b(x - h) + k
Note: Logarithms are the inverses of exponential functions.

1. Exponential Graphs

Basic Shape:

• b > 1 → Increasing
• 0 < b < 1 → Decreasing
• Horizontal Asymptote: y = k

Domain and Range:

• Domain: (-∞, ∞)
• Range: y > k or y < k (depends on reflection)

Example: f(x) = 2 · 3^x - 2

• Horizontal Asymptote: y = -2
• Domain: (-∞, ∞)
• Range: y > -2

2. Logarithmic Graphs

Basic Shape:

• b > 1 → Increasing
• 0 < b < 1 → Decreasing
• Vertical Asymptote: x = h

Domain and Range:

• Domain: x > h
• Range: (-∞, ∞)

Example: f(x) = log(x + 3) + 2

• Vertical Asymptote: x = -3
• Domain: x > -3
• Range: (-∞, ∞)

3. Finding the Domain of Logarithms

Rule: The expression inside the logarithm must be greater than zero (> 0).

Example: y = ln(x - 1)
x - 1 > 0 → x > 1
Domain: (1, ∞)

4. Rewriting Logarithms as Exponentials

Rule: log_b(a) = c ⇔ b^c = a

Examples:

• log_3(12) = 2 → 3^2 = 12
• log_b(x) = 1/2 → b^(1/2) = x

5. Rewriting Exponentials as Logarithms

Rule: b^c = a ⇔ log_b(a) = c

Examples:

• 18^x = 324 → log_18(324) = x
• 17^a = d → log_17(d) = a

6. Logarithmic Properties

• Product: log(a) + log(b) = log(ab)
• Quotient: log(a) - log(b) = log(a/b)
• Power: log(a^n) = n · log(a)

Example: log(12) + log(x) = log(12x)

7. Solving Logarithmic Equations

Method A (Same Base): log(a) = log(b) → a = b

Method B (Single Log):

1. Combine logs.
2. Rewrite as an exponential.
3. Solve.
4. Check the domain.

Example: log(x - 2) = 3
x - 2 = 10^3 → x = 1002

8. Solving Exponential Equations

Method A (Same Base): b^x = b^y → x = y

Method B (Using Logs):

1. Isolate the exponential.
2. Take the log of both sides.
3. Solve.

Example: 5^(x + 1) = 70
log(5^(x + 1)) = log(70)
(x + 1) log 5 = log 70

Final Graph Checklist

• Exponential: Horizontal asymptote.
• Logarithmic: Vertical asymptote.
• Log Domain: Inside expression must be > 0.
• Exponential Domain: All real numbers.
• Relationship: Logs and exponentials are inverses.