H2 Math Topic 1.1 Functions

Singapore-Cambridge H2 Mathematics (9758) | 2027 A-level Examination Syllabus

Learning objectives

By the end of this lesson, you should be able to:

1. Understand the concepts of functions, domain and range.
2. Determine whether a function is one-to-one.
3. Find and use inverse functions.
4. Find and use composite functions.
5. Determine the conditions for inverse and composite functions to exist.
6. Restrict the domain of a function to obtain an inverse.
7. Understand the relationship between the graphs of a function and its inverse.

1. What is a Function?

A function is a rule that assigns exactly one output to every permitted input.

For example, consider:

$$f\left(x\right)=2x+3$$

For each value of x, the function produces exactly one value of f(x).

Input, x	Output, f(x)
-2	-1
-1	1
0	3
1	5
2	7
3	9

Important: Every input must have exactly one output.

• One input cannot produce two different outputs.
• Different inputs are allowed to produce the same output.

Function notation

The notation f(x) represents the output of function f when the input is x.

Suppose:

$$f\left(x\right)=x^2+2x-1$$

Find f(3).

Solution:

$$f\left(3\right)=3^2+2\left(3\right)-1$$ $$=9+6-1=14$$

Answer: f(3) = 14.

2. Domain and Range

The domain is the set of all permitted input values. The range is the set of all output values that the function actually produces.

Domain: All possible values of x.

Range: All possible values of f(x).

Example 1: A quadratic function

Consider:

$$f\left(x\right)=x^2$$

Assume the domain is all real numbers. The square of every real number is non-negative.

$$\text{Domain:}x\in ℝ$$ $$\text{Range:}f\left(x\right)\ge 0$$

Example 2: A reciprocal function

Consider:

$$f\left(x\right)=\frac{1}{x}$$

The denominator cannot be zero, and the output can never be zero.

$$\text{Domain:}x\in ℝ,x\ne 0$$ $$\text{Range:}f\left(x\right)\in ℝ,f\left(x\right)\ne 0$$

Common domain restrictions

Function	Restriction
$\sqrt{x}$	$x\ge 0$
$\frac{1}{x}$	$x\ne 0$
$\ln \left(x\right)$	$x>0$
$\ln (x-2)$	$x>2$

3. One-to-One Functions

A function is called one-to-one if different inputs always produce different outputs.

In other words, if two outputs are equal, their corresponding inputs must also be equal.

$$f\left(a\right)=f\left(b\right)\Rightarrow a=b$$

Example 1: One-to-one

$$f\left(x\right)=2x+1$$

This function is one-to-one because each output corresponds to exactly one input.

Example 2: Not one-to-one

$$f\left(x\right)=x^2$$

Notice that:

$$f\left(2\right)=4=f(-2)$$

Two different inputs produce the same output. Therefore, this function is not one-to-one over all real numbers.

Horizontal line test: A function is one-to-one if every horizontal line intersects its graph at most once.

4. Inverse Functions

An inverse function reverses the operation of the original function.

If a function changes input x into output y, its inverse changes y back into x.

$$f^{-1}\left(f\right(x\left)\right)=x$$

The notation f^-1(x) means the inverse function. It does not mean the reciprocal of f(x).

How to find an inverse function

1. Write y = f(x).
2. Make x the subject of the equation.
3. Interchange x and y.
4. Write the answer as f^-1(x).
5. State the appropriate domain and range.

Worked Example 1

Find the inverse of:

$$f\left(x\right)=3x-5$$

Step 1: Write y = f(x).

$$y=3x-5$$

Step 2: Make x the subject.

$$y+5=3x$$ $$x=\frac{y+5}{3}$$

Step 3: Interchange x and y.

$$y=\frac{x+5}{3}$$

Final answer:

$$f^{-1}\left(x\right)=\frac{x+5}{3}$$

Conditions for an inverse function

For an inverse to be a function, the original function must be one-to-one on its domain.

The inverse function exchanges the domain and range of the original function.

$$\text{Domain of}{f}^{-1}=\text{Range of}f$$ $$\text{Range of}{f}^{-1}=\text{Domain of}f$$

5. Domain Restriction to Obtain an Inverse

Some functions are not one-to-one over their natural domains. However, we can restrict their domains to make them one-to-one.

Worked Example: Quadratic function

Consider:

$$f\left(x\right)=x^2$$

Over all real numbers, this function is not one-to-one. For example:

$$f\left(2\right)=f(-2)=4$$

Therefore, we restrict the domain to:

$$x\ge 0$$

The function is now one-to-one.

Finding the inverse: Step 1: Write y = f(x).

$$y=x^2$$

Step 2: Make x the subject.

$$x=\pm \sqrt{y}$$

However, since the restricted domain is x ≥ 0, we must choose the positive square root.

$$x=\sqrt{y}$$

Step 3: Interchange x and y.

$$y=\sqrt{x}$$

Final answer:

$$f^{-1}\left(x\right)=\sqrt{x}$$

The domain of the inverse is x ≥ 0, and its range is y ≥ 0.

Important examination point: Do not simply find an algebraic inverse. Check whether the original function is one-to-one. If it is not, state the domain restriction needed to make it one-to-one.

6. Composite Functions

A composite function is formed when one function is applied after another function.

The notation is:

$$\mathrm{fg}\left(x\right)=f\left(g\right(x\left)\right)$$

This means that we apply g first, followed by f.

Remember the order:

1. Start with x.
2. Apply g(x).
3. Use the result as the input of f.
4. Obtain fg(x).

Worked Example 1

Given:

$$f\left(x\right)=2x+3$$ $$g\left(x\right)=x^2$$

Find fg(x).

Solution:

$$\mathrm{fg}\left(x\right)=f\left(g\right(x\left)\right)$$ $$=2\left(x^2\right)+3$$ $$=2x^2+3$$

Answer:

$$\mathrm{fg}\left(x\right)=2x^2+3$$

Worked Example 2: Finding a numerical value

Using the same functions, find fg(3).

Step 1: Find g(3).

$$g\left(3\right)=3^2=9$$

Step 2: Apply f to the result.

$$f\left(9\right)=2\left(9\right)+3=21$$

Answer: fg(3) = 21.

Order matters!

In general, fg(x) and gf(x) are not equal.

Using the same functions:

$$\mathrm{fg}\left(x\right)=2x^2+3$$ $$\mathrm{gf}\left(x\right)={(2x+3)}^2$$

These are different expressions.

Conditions for composite functions

For fg(x) to be defined, the output of g must be a valid input for f.

$$\text{Range of g}\subseteq \text{Domain of f}$$

In addition, x must belong to the domain of g. Therefore, the domain of the composite function consists of all permitted inputs of g whose outputs belong to the domain of f.

Worked Example 3: Composite function domain

Suppose:

$$f\left(x\right)=\sqrt{x}$$ $$g\left(x\right)=x-2$$

Find the domain of fg(x).

Step 1: Form the composite function.

$$\mathrm{fg}\left(x\right)=\sqrt{x-2}$$

Step 2: Apply the square root restriction.

$$x-2\ge 0$$

Step 3: Solve.

$$x\ge 2$$

Answer: Domain is x ≥ 2.

7. Relationship Between the Graphs of a Function and Its Inverse

The graph of an inverse function is the reflection of the graph of the original function in the straight line y = x.

This happens because the x- and y-coordinates of every point are interchanged.

If the point (a, b) lies on the graph of y = f(x), then the point (b, a) lies on the graph of y = f^-1(x).

Key properties:

• The domain of f becomes the range of f^-1.
• The range of f becomes the domain of f^-1.
• The two graphs are reflections in the line y = x.
• The intersection points of the two graphs lie on y = x.

Example

Consider:

$$f\left(x\right)=2x+1$$

Its inverse is:

$$f^{-1}\left(x\right)=\frac{x-1}{2}$$

The point (1, 3) on the graph of y = f(x) corresponds to the point (3, 1) on the graph of y = f^-1(x).

Explore the relationship by plotting all three equations in a graphing calculator.

Try these equations in Desmos:

• y = 2x + 1
• y = (x - 1)/2
• y = x

Open Desmos Graphing Calculator

8. Summary of Important Concepts

Concept	Key idea
Function	Each permitted input has exactly one output.
Domain	Set of permitted input values.
Range	Set of actual output values.
One-to-one	Different inputs produce different outputs.
Inverse	Reverses the original function.
Composite	One function is applied after another.
Inverse graph	Reflection in the line y = x.

9. Practice Questions

Try these questions before revealing the answers. They are designed to test your understanding of Topic 1.1.

Question 1: Function notation

Given:

$$f\left(x\right)=2x^2-3x+1$$

Find f(2).

Show answer and working

$$f\left(2\right)=2\left(4\right)-3\left(2\right)+1$$ $$=8-6+1=3$$

Answer: 3.

Question 2: Domain and range

Find the domain and range of:

$$f\left(x\right)=\sqrt{x+4}$$

Show answer and working

For the square root to be defined:

$$x+4\ge 0$$ $$x\ge -4$$

Domain: x ≥ -4. Since a square root is non-negative, the range is f(x) ≥ 0.

Question 3: One-to-one functions

Determine whether the function below is one-to-one over all real numbers.

$$f\left(x\right)=x^2+4$$

Show answer

No. For example:

$$f\left(2\right)=8=f(-2)$$

Different inputs produce the same output.

Question 4: Inverse functions

Find the inverse of:

$$f\left(x\right)=5x+7$$

Show answer and working

Step 1: Write y = f(x).

$$y=5x+7$$

Step 2: Make x the subject.

$$y-7=5x$$ $$x=\frac{y-7}{5}$$

Step 3: Interchange x and y.

$$y=\frac{x-7}{5}$$

Answer:

$$f^{-1}\left(x\right)=\frac{x-7}{5}$$

Question 5: Domain restriction

Given f(x) = x², where x is real, state a suitable domain restriction so that f has an inverse function. Find the inverse.

Show answer

One possible restriction is x ≥ 0.

$$f^{-1}\left(x\right)=\sqrt{x}$$

The other possible restriction is x ≤ 0, which gives:

$$f^{-1}\left(x\right)=-\sqrt{x}$$

Question 6: Composite functions

Given:

$$f\left(x\right)=3x-2$$ $$g\left(x\right)=x+4$$

Find fg(x) and gf(x).

Show answer and working

$$\mathrm{fg}\left(x\right)=3(x+4)-2$$ $$=3x+10$$

$$\mathrm{gf}\left(x\right)=(3x-2)+4$$ $$=3x+2$$

Question 7: Composite function domain

Given:

$$f\left(x\right)=\frac{1}{x}$$ $$g\left(x\right)=x-3$$

Find the domain of fg(x).

Show answer and working

$$\mathrm{fg}\left(x\right)=\frac{1}{x-3}$$

The denominator cannot equal zero.

$$x-3\ne 0$$ $$x\ne 3$$

Domain: all real numbers except 3.

10. Further Learning and Graph Exploration

Graphing is especially useful when studying one-to-one functions, domain restrictions and inverse functions.

Recommended tools:

• Desmos Graphing Calculator – plot and compare functions interactively.
• WolframAlpha – explore functions, inverses and mathematical calculations.

Final examination advice

When answering H2 Mathematics questions on functions, always pay attention to the domain and range. Check whether the function is one-to-one before finding an inverse, and make sure that the range of the inner function is compatible with the domain of the outer function when dealing with composite functions.

Show your mathematical working clearly and state restrictions explicitly.

11. Sources and Further Reading

The following references were used in preparing this lesson. The mathematical explanations and worked examples are independently written.

1. Singapore Examinations and Assessment Board (SEAB).
   Singapore-Cambridge H2 Mathematics (9758), 2027 syllabus, page 6, Topic 1.1 – Functions.
   Official 2027 H2 Mathematics syllabus (PDF)
2. Desmos.
   Getting Started: Desmos Graphing Calculator.
   Desmos Graphing Calculator Guide
3. Wolfram Research.
   WolframAlpha – Computational Knowledge Engine.
   WolframAlpha

This lesson is an independently prepared educational resource based on the Singapore-Cambridge H2 Mathematics (9758) 2027 syllabus. It is not an official SEAB or Cambridge publication.

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