Mathematics (9260) | Functions, graphs and calculus

Welcome to Functions, Graphs, and Calculus!

Hello future mathematician! This chapter is where things get really exciting. We are moving beyond simple equations and starting to look at how different quantities relate to each other in powerful ways. We will learn how to use functions to model processes, how to understand complex graphs, and we’ll introduce the incredible topic of Calculus—the mathematics of change. Don't worry if this sounds intimidating; we will break down every concept step-by-step!

Why is this important? Functions and graphs are the language of science and economics. Calculus is the foundation used by engineers and physicists to calculate speed, acceleration, and flow rates. Mastering this topic gives you serious mathematical power!

Section 1: Understanding Functions

What is a Function?

A function is essentially a mathematical machine. You put an input into the machine, and it processes it using a fixed rule to produce exactly one output.

• The Input values are often called the Domain.
• The Output values are often called the Range.

The key rule is that for every input, there must be only one unique output. Think of a vending machine: you press 'A1' (input), and you get a specific snack (output). If pressing 'A1' sometimes gave you a drink and sometimes gave you a crisp packet, it wouldn't be a consistent function!

Function Notation: \(f(x)\)

Instead of writing \(y = 2x + 1\), we use function notation:

\(f(x) = 2x + 1\)

We read this as "f of x equals 2x plus 1."

How to Evaluate a Function

If you are asked to find \(f(3)\), it means you substitute 3 wherever you see \(x\).

Step-by-Step Example:

1. Start with the function rule: \(f(x) = x^2 - 5\)
2. Substitute the input (4): \(f(4) = (4)^2 - 5\)
3. Calculate: \(f(4) = 16 - 5\)
4. Result: \(f(4) = 11\)

Key Takeaway: \(f(x)\) is just a fancy way of writing \(y\).

Composite Functions: \(fg(x)\)

A composite function is when the output of one function becomes the input for another function. It’s like a chain reaction!

If we have two functions, \(f(x)\) and \(g(x)\), the composite function \(fg(x)\) means: First apply \(g\), then apply \(f\).

Analogy: Imagine two processes in a factory. \(g\) is the first step (cutting the wood), and \(f\) is the second step (painting the wood). The output of the cutting machine goes straight into the painting machine.

Step-by-Step for finding \(fg(x)\):

Let \(f(x) = 2x + 1\) and \(g(x) = x^2\)

1. Identify the 'inner' function: \(g(x) = x^2\).
2. Substitute the entire inner function into the outer function \(f\):

   Instead of \(f(x) = 2(x) + 1\), we write:

   \(fg(x) = f(g(x)) = 2(x^2) + 1\)

3. Simplify: \(fg(x) = 2x^2 + 1\)

Common Mistake Alert! Always work from the inside out. \(fg(x)\) is NOT the same as \(gf(x)\).

Inverse Functions: \(f^{-1}(x)\)

The inverse function, denoted by \(f^{-1}(x)\), is the function that undoes what the original function \(f(x)\) did. If \(f(5) = 10\), then \(f^{-1}(10) = 5\).

The graph of \(f^{-1}(x)\) is always a reflection of the graph of \(f(x)\) across the line \(y = x\).

Step-by-Step for finding \(f^{-1}(x)\):

Let \(f(x) = 3x - 6\)

1. Replace \(f(x)\) with \(y\):

   \(y = 3x - 6\)

2. Swap \(x\) and \(y\): (This is the crucial step that creates the inverse relationship)

   \(x = 3y - 6\)

3. Rearrange the equation to make \(y\) the subject (solve for \(y\)):

   \(x + 6 = 3y\)

   \(\frac{x + 6}{3} = y\)

4. Replace \(y\) with \(f^{-1}(x)\):

   \(f^{-1}(x) = \frac{x + 6}{3}\)

Memory Aid: To find the inverse, Swap and Solve!

Section 2: Advanced Graph Sketching and Transformations

Key Graph Shapes You Must Know

Understanding the basic shape of a function is crucial for sketching graphs.

1. Linear Functions: \(y = mx + c\)

   Shape: A straight line.

2. Quadratic Functions: \(y = ax^2 + bx + c\)

   Shape: A parabola (U-shape or inverted U-shape). If \(a > 0\), it’s a happy face (minimum point). If \(a < 0\), it’s a sad face (maximum point).

3. Cubic Functions: \(y = ax^3 + \dots\)

   Shape: An S-shape, or a curve that levels off briefly. It has up to two turning points.

4. Reciprocal Function: \(y = \frac{k}{x}\)

   Shape: Two separate curves (in opposing quadrants) that never touch the axes. The axes are called asymptotes.

5. Exponential Functions: \(y = k^x\)

   Shape: A curve that grows or decays very rapidly. It crosses the y-axis at \((0, 1)\) (if \(k>0\)) and never touches the x-axis.

Graph Transformations

If you know the graph of a simple function \(y = f(x)\), you can easily find the graph of a related function by shifting, stretching, or reflecting it.

Let's assume we start with the graph \(y = f(x)\).

1. Translation (Shifting)

Translation moves the graph without changing its shape or orientation.

• Vertical Shift (Outside Change):

  \(y = f(x) + a\)

  Effect: Shifts the graph up by \(a\) units (if \(a\) is positive).

  Example: The graph of \(y = x^2 + 3\) is the graph of \(y = x^2\) moved up 3 units.



• Horizontal Shift (Inside Change):

  \(y = f(x + a)\)

  Effect: Shifts the graph left by \(a\) units (if \(a\) is positive).

  IMPORTANT TRICK: Changes inside the bracket affect the x-coordinate, and they are the opposite of what you expect. If you see \(+a\), you move left (negative x direction).

Memory Aid: INSIDE LIES! (Changes inside the function brackets do the opposite of what the sign suggests.)

2. Reflection

Reflection flips the graph across an axis.

• Reflection in the x-axis:

  \(y = -f(x)\)

  Effect: All y-coordinates are multiplied by \(-1\). The graph flips vertically.



• Reflection in the y-axis:

  \(y = f(-x)\)

  Effect: All x-coordinates are multiplied by \(-1\). The graph flips horizontally.

Section 3: Introduction to Calculus: Differentiation

Calculus is the mathematics of change. It allows us to calculate how fast things are changing at any exact moment in time.

The Gradient of a Curve

When dealing with straight lines, the gradient is constant. But for a curve, the gradient is always changing.

The gradient of a curve at a specific point is defined as the gradient of the tangent line at that point.

• A Tangent is a straight line that touches the curve at only one point.
• Differentiation is the mathematical process we use to find the formula for the gradient at any point on the curve.

We use the notation \(\frac{dy}{dx}\) to represent the differentiated function (the gradient function).

Key Term: \(\frac{dy}{dx}\) is called the derivative of \(y\) with respect to \(x\).

The Power Rule for Differentiation

For functions in the form \(y = ax^n\), differentiation is very straightforward. This is the only differentiation rule required at this level.

If \(y = ax^n\), then the derivative \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = n \cdot a \cdot x^{n-1} \]

Step-by-Step Rule:

1. Multiply: Bring the power down and multiply it by the coefficient (\(a\)).
2. Reduce: Reduce the power by 1 (\(n-1\)).

Example Differentiation Practice

Example 1: \(y = 4x^3\)

• Multiply: \(3 \times 4 = 12\)
• Reduce power: \(3 - 1 = 2\)
• Result: \(\frac{dy}{dx} = 12x^2\)

Example 2: \(y = 5x\)

• Remember, this is \(y = 5x^1\).
• Multiply: \(1 \times 5 = 5\)
• Reduce power: \(1 - 1 = 0\), so \(x^0 = 1\).
• Result: \(\frac{dy}{dx} = 5\) (The gradient of a straight line is always the coefficient of x!)

Example 3: \(y = 7\) (A constant number)

• Remember, this is \(y = 7x^0\).
• Multiply: \(0 \times 7 = 0\).
• Result: \(\frac{dy}{dx} = 0\) (A horizontal line has zero gradient.)

Rule for Sums: If a function has multiple terms, you differentiate each term separately:

If \(y = 2x^4 - 3x^2 + 6x - 1\)

\[ \frac{dy}{dx} = 8x^3 - 6x + 6 \]

Prerequisite Skill Check: Sometimes you need to rearrange the expression before differentiating:

• If you see a root: \(\sqrt{x}\) must be written as \(x^{\frac{1}{2}}\).
• If you see \(x\) in the denominator: \(\frac{1}{x^2}\) must be written as \(x^{-2}\).

Applications of Differentiation

1. Finding the Gradient at a Specific Point

If you want to find the gradient of \(y = x^2 - 3x\) at the point where \(x = 5\):

1. Find the derivative: \(\frac{dy}{dx} = 2x - 3\)
2. Substitute \(x=5\) into the derivative:

Gradient = \(2(5) - 3 = 10 - 3 = 7\)

2. The gradient of the curve when \(x=5\) is 7.

2. Finding Stationary (Turning) Points

A stationary point (or turning point) is a location on the curve where the gradient is momentarily zero. These are maximum points (peaks) or minimum points (valleys).

At a stationary point, the condition is always: \(\frac{dy}{dx} = 0\)

Step-by-Step for Finding Turning Points:

Find the stationary points of \(y = x^3 - 12x + 5\).

1. Differentiate: \(\frac{dy}{dx} = 3x^2 - 12\)
2. Set the derivative equal to zero: \(3x^2 - 12 = 0\)
3. Solve for \(x\):

   \(3x^2 = 12\)

   \(x^2 = 4\)

   \(x = 2\) or \(x = -2\)

4. Find the corresponding \(y\) coordinates by substituting these \(x\) values back into the original equation (\(y\)):

   If \(x=2\): \(y = (2)^3 - 12(2) + 5 = 8 - 24 + 5 = -11\). Point: \((2, -11)\)

   If \(x=-2\): \(y = (-2)^3 - 12(-2) + 5 = -8 + 24 + 5 = 21\). Point: \((-2, 21)\)

The stationary points are \((2, -11)\) and \((-2, 21)\).

Note: Determining whether these are maxima or minima usually involves examining the shape of the graph (e.g., cubic positive x³ starts low, ends high) or checking the sign of the gradient just before and after the stationary point.