Evaluating Functions with Notation

Hello! Welcome to your next lesson in the Functions and Sequences module.

In the last lesson, we focused on finding the nth term for linear and quadratic sequences. That was all about understanding a rule that generates a list of numbers. Today, we're going to look at a more general and powerful way of describing mathematical rules, known as function notation. Mastering this is essential as it's the language used in topics like graphs, transformations, and calculus, which are all coming up.

Your goal for this lesson is to understand function notation and use it to evaluate functions for both numerical and algebraic inputs. This is a core skill that appears frequently in exams, and getting it right is key to picking up guaranteed marks.

1. What is Function Notation?

You're already familiar with equations like $y = 2 x + 5$ . Function notation is simply a more formal way to write this. Instead of y, we might write f(x).

So, $y = 2 x + 5$ becomes $f (x) = 2 x + 5$ .

We read f(x) as "f of x". It means "a function named 'f' that depends on the variable 'x'". Think of f as the name of the function or the rule. You'll also see other letters used, like g(x) or h(x).

A great way to visualize this is as a "function machine": you put an input (x) in, the machine applies the rule (the function), and you get an output (f(x)) out.

Introduction to Functions and Function Notation — This diagram shows a simple function 'f' that maps an input 'x' to an output 'x + 4'. The notation f(x) = x + 4 is just a way of writing down this rule.

To see this idea explained, let's watch the beginning of a short video.

Function Notation - GCSE Higher Maths

This video from 1st Class Maths clearly explains the 'function machine' analogy and how it relates to f(x) notation.

Watch from the beginning until 01:40. Focus on how the input 'x' is processed by the function to produce the output '2x + 5', and how this is written as f(x) = 2x + 5.

A Crucial Point on Notation
One of the most common mistakes students make is to think f(x) means f multiplied by x. This is incorrect. The brackets in f(x) do not mean multiplication. It is a special notation meaning "the function f applied to the input x".

2. Evaluating Functions with Numerical Inputs

This is the most direct application of function notation. If you have a function like $f (x) = 3 x - 1$ and you're asked to find f(2), it simply means "find the output when the input is 2". You substitute 2 everywhere you see x.

$f (x) = 3 x - 1$
$f (2) = 3 (2) - 1 = 6 - 1 = 5$

Pro Tip for Avoiding Errors: When you substitute a value, especially a negative one, always put it in brackets. This will help you avoid the sign errors you've mentioned are a weak point.

For example, to find f(-4) for f(x) = x^2 + 10:

Write f(-4) = (-4)^2 + 10
Calculate (-4)^2 = 16, not -16.
So, f(-4) = 16 + 10 = 26.

Let's see this technique applied to a few examples.

Function Notation - GCSE Higher Maths

Continuing with the same video, let's watch how to evaluate functions for different numerical inputs, including negatives.

Watch from 01:40 to 04:49. Notice how for each question, the x is replaced by the input value in brackets. Pay close attention to the g(-3) example at 04:14 to see how to handle negative numbers.

Test your understanding!

Given the function $g (x) = x^{2} - 5 x$ , find the value of $g (- 3)$ .

Show answer

We need to substitute x = -3 into the expression for $g (x)$ . Using brackets is essential here.

$g (- 3) = (- 3)^{2} - 5 (- 3)$
$g (- 3) = 9 - (- 15)$
$g (- 3) = 9 + 15$
$g (- 3) = 24$

3. Evaluating Functions with Algebraic Inputs

The process doesn't change when the input is an algebraic expression, like 2a or x+3. You still substitute the entire expression everywhere you see an x. This is where your algebra skills from previous lessons, like expanding brackets, are very important.

For example, if $f (x) = 5 x + 7$ and we want to find f(2m):

Replace x with (2m).
$f (2 m) = 5 (2 m) + 7 = 10 m + 7$

If $g (x) = x^{2} - 1$ and we want to find g(a+3):

Replace x with (a+3).
$g (a + 3) = (a + 3)^{2} - 1$
Expand the bracket: $(a + 3) (a + 3) = a^{2} + 3 a + 3 a + 9 = a^{2} + 6 a + 9$
Simplify: $g (a + 3) = (a^{2} + 6 a + 9) - 1 = a^{2} + 6 a + 8$

Let's review some worked examples to solidify this process.

Functions | CIE IGCSE Maths: Extended Revision Notes 2025

The revision notes from Save My Exams include a very clear worked example of evaluating a function for both a number and an algebraic expression. This is great for comparison.

Please read the 'Worked example' on page 4 of the PDF. Focus on part (a) finding f(7) and part (b) finding f(x+3). Observe how the method of substitution is identical in both cases.

Now, let's watch a video that demonstrates this with another common type of algebraic input that often appears in calculus.

Evaluating Functions (Intro to Function Notation)

This video from Mario's Math Tutoring shows how to handle an input of x+h. This is a classic example that requires careful expansion and simplification.

Watch the segment from 01:30 to 03:05. Pay close attention to how he substitutes (x+h) into 4x² - x and methodically simplifies the result. This reinforces the importance of careful algebraic manipulation.

4. Working Backwards and Exam Practice

A common exam question asks you to do the reverse: you're given the output and asked to find the input.

For example: Given $f (x) = 4 x - 5$ , find the value of x for which f(x) = 15.

Here, you set the function's rule equal to the given output and solve the resulting equation:

Set up the equation: $4 x - 5 = 15$
Solve for x:
$4 x = 20$
$x = 5$

This turns a function problem into an equation-solving problem, linking back to your core algebra skills.

Now, let's put everything together with some exam-style practice.

Test your understanding! (Exam Practice)

Attempt the following questions, which are typical of IGCSE papers. Pay attention to the marks available to judge how much working is needed.

Question 1
Given that $f (x) = 5 x - 2$ :
(a) Find $f (- 4)$
(b) Find x when f(x) = 8
(3 marks)

Question 2
(a) If $f (x) = 4 x + 1$ , write a simplified expression for f(2a-1)
(b) If $h (x) = x^{2} + 2 x - 3$ , write a simplified expression for h(2m+3)
(5 marks)

Question 3 (Challenge)
Given that $f (x) = x^{2} + 2$ and $g (x) = 3 (x + 4)$ , find the values of x which satisfy f(x) = g(x).
(4 marks)

Show answers and working

These questions are based on the "Function notation GCSE questions" from the Third Space Learning resource.

Answer 1
(a) Substitute x = -4:
$f (- 4) = 5 (- 4) - 2 = - 20 - 2 = - 22$ (1 mark)

(b) Set the function equal to 8:
$5 x - 2 = 8$ (1 mark for the equation)
$5 x = 10$
$x = 2$ (1 mark for the answer)

Answer 2
(a) Substitute x = 2a-1:
$f (2 a - 1) = 4 (2 a - 1) + 1$ (1 mark for substitution)
$= 8 a - 4 + 1$
$= 8a - 3$ (1 mark for simplifying)

(b) Substitute x = 2m+3:
$h (2 m + 3) = (2 m + 3)^{2} + 2 (2 m + 3) - 3$ (1 mark for substitution)
Expand brackets: $(4 m^{2} + 12 m + 9) + (4 m + 6) - 3$ (1 mark for correct expansion)
Simplify by combining like terms: $4 m^{2} + 16m + 12$ (1 mark for final simplified expression)

Answer 3
Set the functions equal to each other:
$x^{2} + 2 = 3 (x + 4)$ (1 mark)
Expand and rearrange into a quadratic equation equal to zero:
$x^{2} + 2 = 3 x + 12$
$x^{2} - 3 x - 10 = 0$ (1 mark)
Factorize the quadratic:
$(x - 5) (x + 2) = 0$ (1 mark)
Solve for the two possible values of x:
$x = 5$ or $x = - 2$ (1 mark for both correct solutions)

Conclusion

Well done. You've now covered the fundamentals of function notation, a concept that will reappear throughout your remaining lessons.

Key Takeaways:

Function notation like f(x) is a formal way to write a mathematical rule, where f is the name and x is the input.
To evaluate a function, you substitute the given input (whether it's a number or an algebraic term) in place of x.
Always use brackets when substituting negative numbers or algebraic expressions to prevent common arithmetic and sign errors.
Finding x for a given f(x) value means you must set up and solve an equation.

Preview of the Next Lesson:
Now that you can work with single functions, we will look at what happens when we combine them. In the next lesson, you'll learn how to find composite functions, written as fg(x), which means applying function g first, and then applying function f to the result. This will use today's substitution skills directly.

Recommended resources

1. What is Function Notation?

2. Evaluating Functions with Numerical Inputs

3. Evaluating Functions with Algebraic Inputs

4. Working Backwards and Exam Practice

Conclusion