CXC/CSEC Mathematics: Functions, Relations, and Graphs

1. Introduction to Relations

A relation is a connection between elements from one set (called the domain) to elements of another set (called the range). Relations can be represented in various ways:

1.1 Relations as Sets of Ordered Pairs

An ordered pair is written as (x, y), where x is from the domain and y is from the range.

Example: The relation R = {(1, 2), (3, 4), (5, 6)} maps elements from set A = {1, 3, 5} to elements in set B = {2, 4, 6}.

Domain of R = {1, 3, 5}

Range of R = {2, 4, 6}

1.2 Mapping Diagrams

A mapping diagram visually shows how elements from the domain are connected to elements in the range.

Domain Range 1 3 5 2 4 6

Figure 1: Mapping diagram of relation R = {(1, 2), (3, 4), (5, 6)}

2. Functions

A function is a special type of relation where each element in the domain is related to exactly one element in the range.

Definition: A function is a relation in which each input value (domain) corresponds to exactly one output value (range).

2.1 Determining if a Relation is a Function

To determine if a relation is a function, we use the vertical line test: If a vertical line intersects the graph of a relation at more than one point, then the relation is not a function.

x y O

This is a function (passes vertical line test)

x y O

This is not a function (fails vertical line test)

2.2 Function Notation

We typically use the notation f(x) to represent a function, where x is the input (independent variable) and f(x) is the output (dependent variable).

Example: For the function f(x) = 2x + 3

If x = 4, then f(4) = 2(4) + 3 = 8 + 3 = 11

2.3 Domain and Range of Functions

For any function:

Example: For f(x) = x² - 4

Domain: All real numbers (R)

Range: y ≥ -4 (or {y ∈ R | y ≥ -4})

3. Types of Functions

3.1 Linear Functions

A linear function has the form f(x) = mx + c, where m is the gradient (slope) and c is the y-intercept.

x y O c = -10

Figure 2: Graph of a linear function f(x) = 2x - 10

3.2 Quadratic Functions

A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola.

x y O Vertex

Figure 3: Graph of a quadratic function f(x) = x² - 4

Important features of a quadratic function:

3.3 Cubic Functions

A cubic function has the form f(x) = ax³ + bx² + cx + d, where a ≠ 0.

x y O

Figure 4: Graph of a cubic function f(x) = x³

3.4 Exponential Functions

An exponential function has the form f(x) = a^x or f(x) = a · b^x where a and b are constants, b > 0, and b ≠ 1.

x y O Asymptote

Figure 5: Graph of an exponential function f(x) = 2^x

4. Graphing Functions and Relations

4.1 Plotting Points

To plot a function or relation:

  1. Create a table of values by substituting x-values into the function
  2. Plot the resulting ordered pairs (x, y) on the coordinate plane
  3. Connect the points with a smooth curve if the function is continuous

Example: Graph f(x) = x² - 2

x f(x) = x² - 2 Point (x, y)
-2 (-2)² - 2 = 4 - 2 = 2 (-2, 2)
-1 (-1)² - 2 = 1 - 2 = -1 (-1, -1)
0 (0)² - 2 = 0 - 2 = -2 (0, -2)
1 (1)² - 2 = 1 - 2 = -1 (1, -1)
2 (2)² - 2 = 4 - 2 = 2 (2, 2)
x y O -2 -1 1 2 2 1 -1 -2

Figure 6: Graph of f(x) = x² - 2

4.2 Finding Intercepts

x-intercepts: Points where the graph crosses the x-axis (y = 0)

y-intercept: Point where the graph crosses the y-axis (x = 0)

Example: Find the intercepts of f(x) = x² - 4

x-intercepts: Set f(x) = 0

x² - 4 = 0

x² = 4

x = ±2

x-intercepts are (-2, 0) and (2, 0)

y-intercept: Calculate f(0)

f(0) = 0² - 4 = -4

y-intercept is (0, -4)

5. Transformation of Functions

Transformations allow us to modify basic functions to create new functions. The main types of transformations are:

5.1 Vertical Translations

Adding or subtracting a constant to the function: f(x) + k or f(x) - k

5.2 Horizontal Translations

Replacing x with (x - h) or (x + h): f(x - h) or f(x + h)

x y O f(x) = x² f(x) = x² + 20 f(x) = (x-20)²

Figure 7: Function transformations

5.3 Reflections

Two common reflections:

5.4 Stretching and Compression

Multiplying the function by a constant: a·f(x)

Replacing x with ax: f(ax)

6. Composite Functions

A composite function is a function formed by combining two or more functions.

Definition: If f and g are functions, the composite function (f ∘ g)(x) is defined as f(g(x)).

Example: If f(x) = x² and g(x) = x + 3, find (f ∘ g)(x) and (g ∘ f)(x).

(f ∘ g)(x) = f(g(x)) = f(x + 3) = (x + 3)² = x² + 6x + 9

(g ∘ f)(x) = g(f(x)) = g(x²) = x² + 3

7. Inverse Functions

An inverse function reverses the effect of a function. If f maps x to y, then the inverse function f⁻¹ maps y back to x.

Definition: A function f has an inverse function f⁻¹ if and only if f is a one-to-one function (injective).

7.1 Finding Inverse Functions (Continued)

  1. Swap x and y: x = f(y)
  2. Solve for y to find y = f⁻¹(x)
  3. Check that the domains and ranges are appropriate

Example: Find the inverse of f(x) = 2x + 3

Step 1: Replace f(x) with y

y = 2x + 3

Step 2: Swap x and y

x = 2y + 3

Step 3: Solve for y

x - 3 = 2y

y = (x - 3)/2

Therefore, f⁻¹(x) = (x - 3)/2

7.2 Graphing Inverse Functions

The graph of f⁻¹ is the reflection of the graph of f about the line y = x.

x y O y = x f(x) = 2x + 3 f⁻¹(x) = (x-3)/2

Figure 8: Graph of a function and its inverse

7.3 Horizontal Line Test

A function has an inverse if and only if it passes the horizontal line test: No horizontal line intersects the graph at more than one point.

A function that passes the horizontal line test is called a one-to-one function or an injective function.

8. Function Applications in Real Life

8.1 Linear Functions in Real Life

Linear functions are widely used to model relationships in real life, such as:

Example: A taxi company charges $5 for pickup plus $2 per kilometer traveled. The cost function C(x) for a journey of x kilometers is:

C(x) = 2x + 5

If your journey is 10 km, the cost will be:

C(10) = 2(10) + 5 = $25

8.2 Quadratic Functions in Real Life

Quadratic functions can model:

Example: A ball is thrown upward from a height of 2 meters with an initial velocity of 20 m/s. The height h(t) of the ball after t seconds is given by:

h(t) = -4.9t² + 20t + 2

The height after 2 seconds is:

h(2) = -4.9(2)² + 20(2) + 2 = -19.6 + 40 + 2 = 22.4 meters

8.3 Exponential Functions in Real Life

Exponential functions can model:

Example: A $1000 investment earns 5% interest compounded annually. The value A(t) after t years is:

A(t) = 1000(1.05)^t

After 10 years, the investment will be worth:

A(10) = 1000(1.05)^10 ≈ $1628.89

9. Solving Graphical Problems

9.1 Finding Points of Intersection

The points of intersection of two graphs represent the solutions to the system of equations.

Example: Find the points of intersection of y = x² - 2 and y = 2x - 1

At the points of intersection, the y-values are equal:

x² - 2 = 2x - 1

x² - 2x - 1 = 0

Using the quadratic formula:

x = (2 ± √(4 + 4))/2 = (2 ± √8)/2 = (2 ± 2√2)/2 = 1 ± √2

So x = 1 + √2 ≈ 2.414 or x = 1 - √2 ≈ -0.414

Substituting back to find the y-coordinates:

When x = 1 + √2:

y = 2(1 + √2) - 1 = 2 + 2√2 - 1 = 1 + 2√2 ≈ 3.83

When x = 1 - √2:

y = 2(1 - √2) - 1 = 2 - 2√2 - 1 = 1 - 2√2 ≈ -1.83

The points of intersection are approximately (2.414, 3.83) and (-0.414, -1.83)

x y O y = x² - 2 y = 2x - 1

Figure 9: Points of intersection of two functions

9.2 Finding Maximum and Minimum Values

For a quadratic function f(x) = ax² + bx + c:

Example: Find the maximum value of f(x) = -2x² + 8x + 3

Since a = -2 < 0, this function has a maximum value.

x-coordinate of vertex: x = -b/(2a) = -8/(2(-2)) = -8/-4 = 2

Maximum value: f(2) = -2(2)² + 8(2) + 3 = -8 + 16 + 3 = 11

The maximum value of the function is 11, occurring at x = 2.

10. Piecewise Functions

A piecewise function is defined by different expressions for different parts of its domain.

Example: Consider the absolute value function:

f(x) = |x| = { x if x ≥ 0 -x if x < 0 }

This is a piecewise function with two pieces.

x y O f(x) = |x|

Figure 10: Graph of the absolute value function f(x) = |x|

11. Self-Assessment Questions

Question 1: Determine whether the following relations are functions. Explain your reasoning.

a) R = {(1, 2), (3, 4), (5, 6)}

b) S = {(1, 2), (1, 3), (2, 4)}

c) T = {(x, y) | y = x²}

a) R is a function because each element in the domain (1, 3, 5) corresponds to exactly one element in the range (2, 4, 6).

b) S is not a function because the element 1 in the domain corresponds to two different elements in the range (2 and 3).

c) T is a function because for each x-value, there is exactly one y-value (y = x²).

Question 2: For the function f(x) = 3x² - 2x + 5, find:

a) f(2)

b) f(-1)

c) The y-intercept

d) The equation of the axis of symmetry

a) f(2) = 3(2)² - 2(2) + 5 = 3(4) - 4 + 5 = 12 - 4 + 5 = 13

b) f(-1) = 3(-1)² - 2(-1) + 5 = 3(1) + 2 + 5 = 3 + 2 + 5 = 10

c) The y-intercept is f(0) = 3(0)² - 2(0) + 5 = 0 + 0 + 5 = 5. So the y-intercept is (0, 5).

d) For a quadratic function f(x) = ax² + bx + c, the axis of symmetry is given by x = -b/(2a).

Here, a = 3, b = -2, so the axis of symmetry is x = -(-2)/(2(3)) = 2/6 = 1/3.

The equation of the axis of symmetry is x = 1/3.

Question 3: Find the domain and range of the following functions:

a) f(x) = √(x - 3)

b) g(x) = 1/(x - 2)

c) h(x) = x² + 4

a) Domain of f(x) = √(x - 3):

Since we can't take the square root of a negative number, we need x - 3 ≥ 0, which means x ≥ 3.

Domain: {x ∈ R | x ≥ 3} or [3, ∞)

Range: Since the square root is always non-negative, the range is [0, ∞).

b) Domain of g(x) = 1/(x - 2):

Since division by zero is undefined, we need x - 2 ≠ 0, which means x ≠ 2.

Domain: {x ∈ R | x ≠ 2} or (-∞, 2) ∪ (2, ∞)

Range: Since a rational function of this form can take any real value except 0, the range is {y ∈ R | y ≠ 0} or (-∞, 0) ∪ (0, ∞).

c) Domain of h(x) = x² + 4:

This function is defined for all real numbers.

Domain: R or (-∞, ∞)

Range: Since x² ≥ 0 for all real numbers, x² + 4 ≥ 4 for all real numbers.

Range: {y ∈ R | y ≥ 4} or [4, ∞)

Question 4: For the functions f(x) = 2x + 1 and g(x) = x² - 3, find:

a) (f ∘ g)(2)

b) (g ∘ f)(2)

c) The expression for (f ∘ g)(x)

d) The expression for (g ∘ f)(x)

a) (f ∘ g)(2) = f(g(2)) = f(2² - 3) = f(4 - 3) = f(1) = 2(1) + 1 = 3

b) (g ∘ f)(2) = g(f(2)) = g(2(2) + 1) = g(5) = 5² - 3 = 25 - 3 = 22

c) (f ∘ g)(x) = f(g(x)) = f(x² - 3) = 2(x² - 3) + 1 = 2x² - 6 + 1 = 2x² - 5

d) (g ∘ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)² - 3 = 4x² + 4x + 1 - 3 = 4x² + 4x - 2

Question 5: Find the inverse of the function f(x) = 3x - 4, if it exists.

To find the inverse of f(x) = 3x - 4:

Step 1: Replace f(x) with y

y = 3x - 4

Step 2: Swap x and y

x = 3y - 4

Step 3: Solve for y

x + 4 = 3y

y = (x + 4)/3

Therefore, f⁻¹(x) = (x + 4)/3

We can verify this by checking that (f ∘ f⁻¹)(x) = (f⁻¹ ∘ f)(x) = x:

(f ∘ f⁻¹)(x) = f((x + 4)/3) = 3((x + 4)/3) - 4 = x + 4 - 4 = x ✓

(f⁻¹ ∘ f)(x) = f⁻¹(3x - 4) = ((3x - 4) + 4)/3 = 3x/3 = x ✓

Question 6: The graph of y = x² is transformed to form the graph of y = -2(x - 3)² + 4. Describe the series of transformations that must be applied.

Starting with y = x²:

1. Replace x with (x - 3): This shifts the graph 3 units to the right, giving y = (x - 3)²

2. Multiply by -2: This stretches the graph vertically by a factor of 2 and reflects it about the x-axis, giving y = -2(x - 3)²

3. Add 4: This shifts the graph 4 units upward, giving y = -2(x - 3)² + 4

In summary: The graph of y = x² is shifted 3 units to the right, stretched vertically by a factor of 2, reflected about the x-axis, and shifted 4 units upward.

12. Glossary of Key Terms

Function: A relation in which each element in the domain corresponds to exactly one element in the range.
Relation: A set of ordered pairs that shows a connection between elements from two sets.
Domain: The set of all input values (x-values) of a function or relation.
Range: The set of all output values (y-values) of a function or relation.
Vertical Line Test: A graphical method to determine if a relation is a function by checking if any vertical line intersects the graph more than once.
Horizontal Line Test: A graphical method to determine if a function is one-to-one by checking if any horizontal line intersects the graph more than once.
Linear Function: A function of the form f(x) = mx + c where m is the slope and c is the y-intercept.
Quadratic Function: A function of the form f(x) = ax² + bx + c where a ≠ 0.
Cubic Function: A function of the form f(x) = ax³ + bx² + cx + d where a ≠ 0.
Exponential Function: A function of the form f(x) = a·b^x where a and b are constants, b > 0, and b ≠ 1.
Composite Function: A function formed by applying one function to the result of another, denoted by (f ∘ g)(x) = f(g(x)).
Inverse Function: A function that reverses the effect of another function, denoted by f⁻¹(x).
One-to-One Function: A function in which each output corresponds to exactly one input (also called injective).
Vertex: The highest or lowest point on the graph of a quadratic function.
Axis of Symmetry: A vertical line passing through the vertex of a parabola, about which the parabola is symmetric.
x-intercept: A point where a graph crosses the x-axis (y = 0).
y-intercept: A point where a graph crosses the y-axis (x = 0).
Transformation: A change in the shape, size, or position of a graph, such as translation, reflection, or scaling.
Piecewise Function: A function defined by different expressions for different parts of its domain.

13. Key Formulas

Quadratic Formula:

For a quadratic equation ax² + bx + c = 0 (a ≠ 0), the solutions are given by:

x = [-b ± √(b² - 4ac)] / (2a)

1. Use the quadratic formula to solve: 2x² + 5x - 3 = 0

a = 2, b = 5, c = -3

x = [-5 ± √(5² - 4·2·(-3))] / (2·2)

x = [-5 ± √(25 + 24)] / 4

x = [-5 ± √49] / 4

x = [-5 ± 7] / 4

Solutions: x = (-5 + 7)/4 = 0.5 and x = (-5 - 7)/4 = -3

2. What is the discriminant of 3x² - 4x + 1 = 0 and what does it tell us about the roots?

Discriminant D = b² - 4ac = (-4)² - 4·3·1 = 16 - 12 = 4

Since D > 0, there are two distinct real roots.

3. Find the y-intercept of the function f(x) = 2x² - 8x + 6

The y-intercept occurs when x = 0:

f(0) = 2(0)² - 8(0) + 6 = 6

y-intercept is at (0, 6)