1 Piecewise Functions
A piecewise function is defined by different rules for different parts of its domain.
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Notation: f(x) = { rule₁ if condition₁ ; rule₂ if condition₂ }
Continuous vs Discontinuous
| Type | Definition |
|---|---|
| Continuous | No jumps or breaks — you can draw it without lifting your pencil. |
| Discontinuous | Has breaks or jumps at boundary points between pieces. |
Endpoints: Open vs Closed Dots
- Closed dot (●): the point is included — inequality uses ≤ or ≥.
- Open dot (○): the point is excluded — inequality uses < or >.
- Each x-value can only have ONE y-value (function rule still applies).
Graphing Steps
- Step 1: Look at the function — identify how many pieces and what type each piece is (linear, quadratic, etc.).
- Step 2: Make a table for each piece (include the boundary points + at least one middle point).
- Step 3: Mark whether boundary endpoints are open or closed.
- Step 4: Graph each piece separately, using open/closed dots at boundaries.
- Step 5: Do NOT connect pieces across boundaries.
2 Inverse Functions
The inverse of a function f, written f⁻¹, undoes the operation of f. It swaps the roles of x and y.
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Key fact: Domain of f⁻¹ = Range of f · Range of f⁻¹ = Domain of f
When Does an Inverse Exist as a Function?
- The inverse is a function if and only if f is one-to-one (each y value comes from only one x).
- A many-to-one function (like a parabola) has an inverse relation but not a function — unless you restrict the domain.
- Graphical test: original function must pass the horizontal line test.
How to Find the Inverse — 4 Steps
- Step 0: Replace f(x) with y.
- Step 1: Swap x and y.
- Step 2: Solve for y.
- Step 3: Restrict the domain if needed (sketch to check).
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Example — Linear: f(x) = 2x − 6
y = 2x − 6 → swap → x = 2y − 6 → x + 6 = 2y → f⁻¹(x) = (x + 6)/2
y = 2x − 6 → swap → x = 2y − 6 → x + 6 = 2y → f⁻¹(x) = (x + 6)/2
Inverse by Function Type
| Original function | Inverse type | Note |
|---|---|---|
| Linear y = mx + b | Linear | Always a function (as long as m ≠ 0) |
| Quadratic y = a(x−h)² + k | Sideways parabola | NOT a function unless domain restricted; result is √ form |
| Square root y = a√(b(x−h)) + k | Quadratic (restricted domain) | Domain of f⁻¹ = range of f |
| Absolute value y = a|x−h| + k | Two linear pieces (restricted) | Restrict domain of f⁻¹ to [k, +∞) if a > 0 |
Key Properties
- f(f⁻¹(x)) = x and f⁻¹(f(x)) = x — they cancel each other.
- Graphs of f and f⁻¹ are reflections of each other across the line y = x.
- Any point (x, y) on f becomes the point (y, x) on f⁻¹.
- A vertical asymptote on f becomes a horizontal asymptote on f⁻¹.
3 Absolute Value Functions
The absolute value of a number is its distance from zero — always ≥ 0.
|x|
= x if x ≥ 0, = −x if x < 0
|x| = c (c > 0)
x = c OR x = −c (two solutions)
|x| = 0
x = 0 (one solution)
|x| = −c
No solution (absolute value is never negative)
Solving Absolute Value Equations
- Isolate the absolute value expression first.
- Then split into two cases: expression = c and expression = −c.
- Solve each case separately.
- Check both solutions in the original equation.
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Example: |3x − 1| = 26
Case 1: 3x − 1 = 26 → x = 9
Case 2: 3x − 1 = −26 → x = −25/3
Case 1: 3x − 1 = 26 → x = 9
Case 2: 3x − 1 = −26 → x = −25/3
Absolute Value Inequalities
| Form | Equivalent compound inequality | Graph |
|---|---|---|
| |x| < c | −c < x < c (AND) | Bounded region between −c and c |
| |x| > c | x > c OR x < −c | Two separate rays |
| |x| ≤ c | −c ≤ x ≤ c | Closed bounded region |
| |x| ≥ c | x ≥ c OR x ≤ −c | Two closed rays |
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Sign flip: If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the inequality sign.
4 Rational Functions (Overview)
A rational function has the form f(x) = P(x)/Q(x) where P and Q are polynomials and Q(x) ≠ 0.
Standard form (Sec 5)
f(x) = a/(b(x − h)) + k
Vertical asymptote
x = h (where denominator = 0)
Horizontal asymptote
y = k (value function approaches as x → ±∞)
Domain
All real numbers EXCEPT x = h
Range
All real numbers EXCEPT y = k
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The graph consists of two branches — one in each region separated by the vertical asymptote. Finding the inverse swaps the asymptotes.
5 Common Mistakes
| Mistake | Fix |
|---|---|
| Reading piecewise incorrectly at boundaries | Check whether the boundary inequality is < (open) or ≤ (closed) — this changes the dot and which piece "owns" the point. |
| Forgetting to swap BOTH x and y when finding inverse | The entire process is x ↔ y. Swap first, solve second. |
| Assuming every inverse is a function | Check: is the original function one-to-one? If not, restrict the domain. |
| Only finding one solution for |x| = c | There are always two solutions (unless c = 0). Always write both cases. |
| Forgetting to flip inequality when dividing by negative | −4|x| > −8 → |x| < 2 (sign flips!) |