Introduction to Functions and Graphs

Q: What is a function in maths?

A function takes each input and gives exactly one output. You write it as f(x), where x is the input and the value after the equals sign is the output. For example, f(x) = 2x + 1 turns an input of 1 into an output of 3.

Q: What does f(x) mean?

In f(x), the letter f stands for the function and x stands for the input. Whatever comes after the equals sign is the output. So for f(x) = 2x + 1, putting in x = 2 gives f(2) = 5, which means the output is 5.

Q: How do you know if a graph represents a function?

Check whether any input has more than one output. If a single x-value lines up with more than one y-value, the graph is not a function. A function must give exactly one output for each input, so every input can match only one output.

Q: Can one output come from more than one input?

Yes. One output value can come from several different inputs, and the relationship is still a function. The defining feature of a function is that each input has exactly one output, not that each output has only one input, so repeated outputs are fine.

Key concept

A function takes each input and turns it into exactly one output. Put x = 3 into the rule f(x) = 2x + 1 to get f(3) = 7. Show it with a table of values, then plot the points as a graph.

Introduction to Functions and Graphs - introduction visual

Video Lesson

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$Introduction to Functions and Graphs poster$

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Flashcards

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Visual representation of function notation showing the function f(x) = 2x + 1 and its calculated outputs for f(1), f(2), and f(3).

Steps to graph a function showing a table of values for f(x)=2x+1 and its corresponding plotted graph with points.

2 graphs showing non-functions: one with a circle and another with a vertical line, explaining that one x-value must not have multiple y-values.

Graph showing function values with examples: f(-2) = -3 and f(x) = 0 at x = -4, 0, 5, with a function definition.

What Does Function Notation f(x) Mean?

$f (x)$ shows the output when the input is x.
To find $f (3)$ , substitute $x = 3$ into the rule.
For example, if $f (x) = 2 x + 1$ , then $f (3) = 2 (3) + 1 = 7$ .

Practice Questions

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Q1Easy

What is the output of the function $f (x) = 3 x + 2$ when $x = 5$ ?

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Interactive Activity

Introduction to functions and graphs

Plotting

Function Table

Input	Function	Output
0	2 × 0 + 1	1
1	2 × 1 + 1	3
2	2 × 2 + 1	5
3	2 × 3 + 1	7
4	2 × 4 + 1	9
5	2 × 5 + 1	11

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Students Also Ask

The questions students bump into most on this topic

What is a function in maths?

A function takes each input and gives exactly one output. You write it as f(x), where x is the input and the value after the equals sign is the output. For example, f(x) = 2x + 1 turns an input of 1 into an output of 3.

What does f(x) mean?

In f(x), the letter f stands for the function and x stands for the input. Whatever comes after the equals sign is the output. So for f(x) = 2x + 1, putting in x = 2 gives f(2) = 5, which means the output is 5.

How do you know if a graph represents a function?

Check whether any input has more than one output. If a single x-value lines up with more than one y-value, the graph is not a function. A function must give exactly one output for each input, so every input can match only one output.

Can one output come from more than one input?

Yes. One output value can come from several different inputs, and the relationship is still a function. The defining feature of a function is that each input has exactly one output, not that each output has only one input, so repeated outputs are fine.

How do you find the value of a function from a graph?

Remember that y is the output of f(x). To find f(-2), locate x = -2 on the horizontal axis, move to the curve, then read the matching y-value. To solve f(x) = 0, find the inputs where the graph crosses the x-axis.

What is a function machine?

A function machine is a simple way to picture a function. You put a number in as the input, the machine applies its rule, and one number comes out as the output. For example, the rule 2x + 1 turns an input of 1 into an output of 3.

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