Functions Notation: A Revision Guide for GCSE Maths

📚 What is Function Notation?

Function notation is a way of writing mathematical functions using symbols. Instead of writing an equation like $y = 2x + 3$, we use the notation $f(x) = 2x + 3$. This tells us that the function is named "f" and it takes "x" as an input.

• ➡️ Function Name: The letter representing the function (e.g., $f$, $g$, $h$).
• 🔤 Input: The variable inside the parentheses (e.g., $x$, $t$). This represents the value you are plugging into the function.
• 📈 Output: $f(x)$ represents the output or the value of the function for a given input. It's essentially the same as 'y' in a regular equation.

📜 A Brief History

The development of function notation is attributed to mathematicians like Leonhard Euler in the 18th century. His work helped standardize mathematical notation, making it easier to communicate mathematical ideas. Before Euler, different mathematicians used different notations, which made understanding and collaboration difficult. Standardized notation, including function notation, streamlined mathematical discourse and facilitated further advancements.

🔑 Key Principles of Function Notation

• 🔢 Substitution: To evaluate a function at a specific value, substitute that value for the input variable. For example, to find $f(2)$ when $f(x) = 2x + 3$, replace every 'x' with '2'.
• ⚖️ Order of Operations: Follow the order of operations (PEMDAS/BODMAS) when evaluating a function.
• 🎯 Understanding $f(x)$: Remember that $f(x)$ is a single value representing the output of the function.

🌍 Real-World Examples

Function notation isn't just abstract math; it has real-world applications!

• ⛽ Fuel Consumption: Let $C(d)$ be the amount of fuel (in litres) your car uses to drive $d$ kilometers. If $C(100) = 8$, it means your car uses 8 litres of fuel to drive 100 kilometers.
• 🌡️ Temperature Conversion: The function $F(C) = \frac{9}{5}C + 32$ converts Celsius ($C$) to Fahrenheit ($F$).
• 📦 Manufacturing Costs: A factory's cost to produce $n$ items is given by $P(n) = 5n + 1000$. This means each item costs \$5 to make, plus a fixed cost of \$1000 for the factory itself.

✍️ Practice Quiz

Test your understanding with these problems:

1. If $f(x) = 3x - 5$, find $f(4)$.
2. If $g(x) = x^2 + 2x - 1$, find $g(-2)$.
3. If $h(x) = \frac{2x + 1}{x - 3}$, find $h(5)$.
4. If $p(x) = \sqrt{x + 6}$, find $p(3)$.
5. If $q(x) = |2x - 7|$, find $q(1)$.
6. If $r(x) = 4$, find $r(10)$.
7. If $s(x) = x^3$, find $s(-1)$.

Answers:

1. 7
2. -1
3. $\frac{11}{2}$
4. 3
5. 5
6. 4
7. -1

✅ Conclusion

Function notation might seem confusing at first, but with practice, it becomes a powerful tool for representing and working with mathematical relationships. Remember to focus on substitution and understanding what each part of the notation means. Good luck! 👍