Functions (Copy)
FUNCTIONS, COMPOSITE FUNCTIONS, MODULUS FUNCTIONS, GRAPHS AND INVERSE FUNCTIONS
Definition of a Function
A function is a rule that assigns every input value from the domain to exactly ONE output value in the range.
Important Terms
-
Domain → all possible x-values
-
Range → all possible y-values
-
Codomain → target set of outputs
-
Function notation → f(x)
Example
If:
f(x) = x + 2
Then:
-
f(3) = 5
-
f(−1) = 1
Real-Life Example of a Function
A vending machine works like a function.
-
Input → button pressed
-
Output → snack received
Each button gives exactly one snack.
Mapping Diagrams
Mapping diagrams visually show relationships between domain and range.
Example
Domain:
{Alice, Bob, Carol}
Range:
{Sandwich1, Sandwich2, Sandwich3}
Mappings:
-
Alice → Sandwich1
-
Bob → Sandwich2
-
Carol → Sandwich3
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
Identifying Functions
A relation is a function if each input has ONLY ONE output.
Vertical Line Test
A graph represents a function if no vertical line intersects the graph more than once.
Example
y = x² passes the vertical line test.
Therefore, it is a function.
Domain and Range
Domain
All possible x-values that can be used.
Restrictions
-
Denominator cannot equal zero
-
Square roots cannot contain negatives
Example
For:
f(x) = 1/x
Domain
x ≠ 0
Range
y ≠ 0
Piecewise Functions
Functions defined differently in different regions.
Example
f(x) = {
x² if x ≥ 0
−x if x < 0
}
Values
-
f(2) = 4
-
f(−3) = 3
Composite Functions
Composite functions combine two functions.
Definition
(f ∘ g)(x) = f(g(x))
Apply g first, THEN apply f.
Example
Given:
-
g(x) = 2x
-
f(x) = x + 3
Find:
(f ∘ g)(x)
Solution
= f(2x)
= 2x + 3
Order Matters
Generally:
(f ∘ g)(x) ≠ (g ∘ f)(x)
Numerical Example
Given:
-
f(x) = x + 1
-
g(x) = x²
Find:
(f ∘ g)(2)
Step 1
g(2) = 2² = 4
Step 2
f(4) = 5
Final Answer
(f ∘ g)(2) = 5
Symbolic Example
Given:
-
f(x) = 2x − 1
-
g(x) = x² + 3
Find:
(f ∘ g)(x)
Solution
= f(x² + 3)
= 2(x² + 3) − 1
= 2x² + 6 − 1
= 2x² + 5
Conditions for Composite Functions
The range of the first function must fit into the domain of the second function.
Example
Given:
-
g(x) = x²
-
f(x) = √x
Then:
(f ∘ g)(x)
= √(x²)
= |x|
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
Inverse Composite Functions
If both functions are invertible:
(f ∘ g)⁻¹(x) = g⁻¹(f⁻¹(x))
Modulus Functions
Definition
|x| represents distance from zero.
Distance is ALWAYS positive.
Piecewise Definition
|x| = {
x if x ≥ 0
−x if x < 0
}
Properties of Modulus Functions
Symmetry
|x| = |−x|
Therefore modulus functions are EVEN functions.
Graph of |x|
-
V-shaped graph
-
Vertex at (0, 0)
Lines:
-
y = x for x ≥ 0
-
y = −x for x < 0
Solving Modulus Equations
Basic Form
|x| = k
Solutions:
x = k or x = −k
Example
Solve:
|x| = 5
Answer
x = 5 or x = −5
Example 2
Solve:
|2x − 3| = 7
Case 1
2x − 3 = 7
2x = 10
x = 5
Case 2
2x − 3 = −7
2x = −4
x = −2
Final Answers
x = 5, −2
Complex Modulus Equations
For:
|f(x)| = g(x)
Break into TWO cases:
-
f(x) = g(x)
-
f(x) = −g(x)
Example
Solve:
|x − 2| = x + 1
Case 1
x − 2 = x + 1
−2 = 1
No solution
Case 2
x − 2 = −(x + 1)
x − 2 = −x − 1
2x = 1
x = 0.5
Final Answer
x = 0.5
Graphical Representation of Modulus
Transformations
Vertical Shift
y = |x| + c
Shifts graph UP by c units.
Horizontal Shift
y = |x − c|
Shifts graph RIGHT by c units.
Reflection
y = −|x|
Reflection in x-axis.
Stretching
y = a|x|
-
|a| > 1 → vertical stretch
-
0 < |a| < 1 → vertical compression
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
Solving Modulus Graphically
To solve:
|f(x)| = g(x)
-
Draw y = |f(x)|
-
Draw y = g(x)
-
Find intersection points
Example
Solve:
|3x + 4| = 7
Case 1
3x + 4 = 7
3x = 3
x = 1
Case 2
3x + 4 = −7
3x = −11
x = −11/3
Final Answers
x = 1, −11/3
Graphs of Functions
The graph of y = f(x) is the set of all points (x, y).
Important Graph Features
Domain
All allowed x-values
Range
All resulting y-values
Intercepts
-
x-intercepts → where y = 0
-
y-intercept → where x = 0
Symmetry
Shows reflection properties.
Linear Functions
General Form
y = mx + c
Where:
-
m = gradient
-
c = y-intercept
Example
y = 2x + 3
Features
-
Gradient = 2
-
y-intercept = 3
Quadratic Functions
General Form
y = ax² + bx + c
Shape
Parabola
-
a > 0 → opens upward
-
a < 0 → opens downward
Axis of Symmetry
x = −b/2a
Example
y = x² − 4x + 3
Vertex
x = −(−4)/2(1)
= 2
y-coordinate
f(2)
= 2² − 4(2) + 3
= 4 − 8 + 3
= −1
Vertex:
(2, −1)
x-intercepts
x² − 4x + 3 = 0
(x − 1)(x − 3) = 0
x = 1, 3
Cubic Functions
General Form
y = ax³ + bx² + cx + d
S-shaped graph.
Absolute Value Functions
Example
y = |x − 2|
Vertex
(2, 0)
Exponential Functions
General Form
y = aˣ
Where:
-
a > 0
-
a ≠ 1
Types
-
a > 1 → exponential growth
-
0 < a < 1 → exponential decay
Example
y = 2ˣ
Logarithmic Functions
General Form
y = logₐ(x)
Inverse of exponential functions.
Important Feature
Vertical asymptote:
x = 0
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
Transformations of Graphs
Vertical Shifts
-
y = f(x) + k → shift UP
-
y = f(x) − k → shift DOWN
Horizontal Shifts
-
y = f(x + h) → shift LEFT
-
y = f(x − h) → shift RIGHT
Reflections
-
y = −f(x) → reflection in x-axis
-
y = f(−x) → reflection in y-axis
Stretching and Compression
Vertical
y = af(x)
-
|a| > 1 → stretch
-
0 < |a| < 1 → compression
Horizontal
y = f(bx)
-
|b| > 1 → compression
-
|b| < 1 → stretch
Even and Odd Functions
Even Functions
f(−x) = f(x)
Symmetric about y-axis.
Example
f(x) = x²
Odd Functions
f(−x) = −f(x)
Symmetric about origin.
Example
f(x) = x³
Periodic Functions
Repeat values after fixed intervals.
Example
f(x) = sin(x)
Period:
2π
Inverse Functions
Definition
Inverse functions reverse the effect of the original function.
If:
y = f(x)
Then:
-
f(f⁻¹(x)) = x
-
f⁻¹(f(x)) = x
Conditions for Inverse Functions
A function must be ONE-TO-ONE.
Horizontal Line Test
If a horizontal line cuts the graph more than once, the function does NOT have an inverse.
Finding Inverse Functions
Steps
-
Write y = f(x)
-
Swap x and y
-
Solve for y
-
Replace y with f⁻¹(x)
Example
Given:
f(x) = 3x − 7
Step 1
y = 3x − 7
Step 2
x = 3y − 7
Step 3
x + 7 = 3y
y = (x + 7)/3
Final Answer
f⁻¹(x) = (x + 7)/3
Graphs of Functions and Their Inverses
Reflection Property
The graph of f(x) and f⁻¹(x) are reflections in:
y = x
Key Rule
If:
(a, b) lies on f(x)
Then:
(b, a) lies on f⁻¹(x)
Example
For:
f(x) = 2x + 3
Point on function:
(1, 5)
Point on inverse:
(5, 1)
Sketching an Inverse Graph
Steps
-
Draw original graph
-
Draw line y = x
-
Reflect points
-
Join smoothly
Domain and Range Swap
For inverses:
-
Domain of f(x) becomes range of f⁻¹(x)
-
Range of f(x) becomes domain of f⁻¹(x)
Quadratic Inverses
Example:
f(x) = x²
NOT invertible over all real numbers.
Restrict domain:
x ≥ 0
Then:
f⁻¹(x) = √x
Exponential and Logarithmic Inverses
-
y = aˣ
-
y = logₐ(x)
These are inverses.
Trigonometric Inverses
Trig functions require restricted domains.
Example
sin(x)
Restricted domain:
−π/2 ≤ x ≤ π/2
Inverse:
sin⁻¹(x)
Written and Compiled By Sir Hunain Zia (AYLOTI), World Record Holder With 154 Total A Grades, 11 World Records and 7 Distinctions, Educate A Change.
Verifying Inverse Functions
To verify inverses:
-
f(f⁻¹(x)) = x
-
f⁻¹(f(x)) = x
Example
Given:
-
f(x) = 2x − 3
-
f⁻¹(x) = (x + 3)/2
Check 1
f(f⁻¹(x))
= 2[(x + 3)/2] − 3
= x + 3 − 3
= x
Check 2
f⁻¹(f(x))
= [(2x − 3) + 3]/2
= 2x/2
= x
Verified.
Applications of Inverse Functions
Solving Equations
Given:
f(x) = 2x + 1
Solve:
f(x) = 5
Using Inverse
f⁻¹(5)
= (5 − 1)/2
= 2
Real-Life Applications
Unit Conversion
-
Celsius ↔ Fahrenheit
-
km ↔ miles
Cryptography
Encryption and decryption functions behave as inverses.
Practice Questions
-
Find the inverse of:
f(x) = 5x + 7
-
Verify that:
-
f(x) = x³
-
f⁻¹(x) = ∛x
are inverses.
-
Solve:
|3x + 4| = 7
-
Sketch:
y = |x − 2| + 3
-
Solve:
x² − 4x + 3 = 0
-
Sketch:
-
y = x²
-
y = √x
-
y = x
showing inverse symmetry.