Exploring Functions

What Are Functions?

A function is like a special machine in maths. You put something in (an input), the machine does some magic (a rule), and you get something out (an output).

For example:

• The function rule might be “multiply by 2 and then add 3.”
• If you put in $5$, the output will be $(5 \times 2) + 3 = 13$

We often write functions like this: $$f(x) = 2x + 3$$

Here:

• $f(x)$ means "the function of $x$."
• $x$ is the input
• $2x + 3$ is the rule that the function applies.

 

How to Use $f(x)$:

1. Replace $x$ with the input value
2. Follow the function rule to calculate the output.

 

Key Properties of Functions

• Domain: The set of all possible input values.
• Range: The set of all possible output values.

 

Examples

Example 1: $f(x) = 2x + 3$ when $x = 4$:

1. Replace $x$ with $4$ in the rule $f(x) = 2x + 3$: $$f(4) = 2(4) + 3$$
2. Multiply: $$f(4) = 8 + 3$$
3. Add: $$f(4) = 11$$

So, the output is $11$

 

Example 2: $f(x) = 17$ for $f(x) = 2x + 3$:

1. Set the function equal to $17$: $$2x + 3 = 17$$
2. Subtract $3$ from both sides: $$2x = 14$$
3. Divide both sides by $2$: $$x = 7$$

So, the input $x = 7$ gives an output of $17$