Functions

Problem — How to define, represent and use functions to model various situations in 9th grade?

Objectives

Understand the concept of a function and the associated vocabulary: image, antecedent, domain.
Calculate the image of a number using a formula, a table, or a graph.
Determine one or more antecedents of a number.
Graphically represent a simple function, especially an affine function.
Read and interpret the variations of a function.

Part 1: Concept of a function

Important definition

A function associates to each number x in a given set, called the domain, a unique number noted f(x). This number is called the image of x by the function f.

Vocabulary

Antecedent: starting number, the value of x.
Image: obtained result, the value of f(x).
Domain: the set of all values of x for which the function is defined.

Example

For the function f(x) = 2x + 3:

f(4) = 2 × 4 + 3
f(4) = 11

We say that 11 is the image of 4 by function f. We can also say that 4 is an antecedent of 11 for this function.

Domain: forbidden values

If a formula includes a division, you cannot divide by 0.
If a formula includes a square root, you cannot take the square root of a negative number in middle school context.

Examples

f(x)=1/(x-2): this function is not defined for x=2 because it would cause division by zero. The domain is all real numbers except 2.
g(x)=√(x-1): this function is defined only if x-1 ≥ 0, that is, if x ≥ 1.

Important note

In a function, for each value of x, only one image can be associated. This is a fundamental rule. Graphically, this means that if a vertical line crosses a curve in multiple points, then this curve does not represent a function.

Summary of part 1

A function is a very precise mathematical relation: to each allowed value of x, it associates a single output value noted f(x). To work well with functions, you must master the basic vocabulary: image, antecedent, and domain. You must also be able to identify forbidden values in some expressions. This first step is essential as it forms the basis for all calculations, graphical representations, and study of variations in the rest of the course.

Part 2: Calculating images and antecedents

1. Calculating an image

To calculate the image of a number by a function given by a formula, we replace x by the chosen value, then perform the calculations in the usual order.

Example: if f(x) = 3x − 5, then:

f(2) = 3 × 2 − 5
f(2) = 6 − 5 = 1

2. Calculating an antecedent

To find an antecedent of a number b, we look for the value(s) of x that satisfy the equation f(x) = b. This means solving an equation.

Example: find the antecedent of 4 for f(x) = 3x − 5:

3x − 5 = 4
3x = 4 + 5
3x = 9
x = 9 ÷ 3 = 3

So, 3 is an antecedent of 4 by the function f.

Note — Depending on the function studied, the equation f(x)=b may have no solution, one solution, or several solutions. This means a number can have 0, 1, or multiple antecedents.

x	f(x) = 3x − 5
0	−5
2	1
3	4
5	10

Summary of part 2

Calculating an image and finding an antecedent are two different processes. For an image, we start from a value of x and directly compute the result. For an antecedent, we start from a result and must find the corresponding starting value(s). These skills are fundamental in 9th grade because they help understand the meaning of a function, connect calculation and equations, and relate to reading tables or graphs.

Part 3: Graphical representation

Reminder

The graph of a function is the set of points with coordinates (x ; f(x)).
We can create this graph by calculating several images and then plotting the corresponding points.
For an affine function, the graph is a line.

Reading an image on a graph

To read f(a) on a graph:

Locate a on the horizontal axis.
Draw mentally or with a ruler a vertical line up to the curve.
Read the value obtained on the vertical axis: this is the image of a.

Reading an antecedent on a graph

To find the antecedents of b:

Locate b on the vertical axis.
Draw a horizontal line to the curve.
Read the corresponding value(s) of x on the horizontal axis.

Example

To plot the line with equation y = 2x + 1, we can start by calculating some values in a table.

Tip — For an affine function y=ax+b, you can quickly place the point (0 ; b), then a second point like (1 ; a+b).

x	y = 2x + 1
0	1
1	3
2	5
−1	−1

Summary of part 3

The graphical representation of a function helps visualize its behavior. It often provides approximate values but is very useful for quickly reading images, antecedents, or understanding the evolution of a situation. For affine functions, the curve is a line, making their study simpler. Being able to move from a formula to a table, and then from a table to a graph, is an essential skill in the 9th grade curriculum.

Part 4: Study of variations

Definitions

f is increasing if, when x increases, f(x) increases.
f is decreasing if, when x increases, f(x) decreases.
f is constant if the value of f(x) does not change when x varies.

Case of affine functions

If f(x) = ax + b with a > 0, then f is increasing.
If f(x) = ax + b with a < 0, then f is decreasing.
If a = 0, then f(x)=b and the function is constant.

Affine function	Sign of a	Variation
f(x) = ax + b	a > 0	increasing
f(x) = ax + b	a < 0	decreasing
f(x) = ax + b	a = 0	constant

Reading variations on a graph

To study variations, look at the curve from left to right:

if it rises, the function is increasing;
if it falls, the function is decreasing;
if it remains horizontal, the function is constant.

Summary of part 4

Variations describe how a function changes as the value of x changes. They allow us to know if a quantity increases, decreases, or stays stable. In the 9th grade, this concept is particularly important to interpret a graph and understand affine functions. The coefficient a directly gives the direction of change: positive for an increasing function, negative for a decreasing function, zero for a constant function. This reading gives meaning to mathematical models used in real situations.

Final course summary

A function associates a unique image to each allowed number. In 9th grade, you need to recognize this notion, use the right vocabulary, calculate images, determine antecedents, graphically represent a function, and interpret its variations. Affine functions hold an important place because they enable simple modeling of many situations: price evolution, distance traveled, temperature, consumption, or speed. Mastering functions means acquiring an essential tool to link calculation, graphical reading, and problem-solving.