Linear Functions (Economics)

Functions

A function is like a machine that takes a number (an input), changes this number (by adding another number to it, multiplying it by another number etc.) and gives back the changed number (an output). The output is always related to the input in some way so the same input will always give the same output.

We typically denote the input (or independent variable) by $x$ and the output corresponding to $x$ (or the dependent variable) by $y$ or $f(x)$. $f(x)$ says that $f$ is a function of $x$. For example, $x$ might be a person's income and $f(x)$ might be the amount they spend on food each week.

Note: An equation can be called a function when the equation contains only two variables and for each value of one variable there can only be one possible value of the other.

Note: Although the output of a function is commonly denoted by $f(x)$, $f$ can also be replaced by other letters; for example we could use $g(x)$.

The function \[y=5x^2+1\] says that for any input $x$, we obtain the corresponding output $y$ by first squaring $x$, then multiplying by $5$, and finally adding $1$. If we chose $x=3$, then the output would be: \begin{align} y&=5\times(3^2)+1\\ &=5\times9+1\\ &=45+1\\ &=46 \end{align}

Domain and Range

The domain of a function, denoted by the letter $D$, is the set of all possible input values for which the function is defined. It is often useful to draw a sketch to help identify the domain of a particular function.

The range of a function is the set of output values that correspond with a given set of inputs; the range therefore follows from the domain of a function. The range is denoted by the letter $R$.

For more information and examples of finding the domain and range, see Domain and Range.

Linear Functions

A linear function is a function of the general form \[y = ax + b\] where $a$ and $b$ are constants. These functions are called linear as the variable $x$ is always to the power of one.

Examples of functions in economics which are often linear include the consumption function and demand and supply functions.

Linear functions are examples of one-to-one functions. This output of one-to-one functions is unique, which means that each output can only be produced from a single input. We say that there is a one-to-one mapping between the output and the input.

The roots of a linear function are the values of $x$ which make the right-hand side of the function equal to zero. That is, the value of $x$ which satisfy: \[ax+b=0\] All linear functions have only one root.

The graph of a linear function is always a straight line where $a$ is the slope or gradient of the line, and $b$ represents the $y$-axis intercept. The $x$-axis intercept of a linear function is the root of the function.

The best way to plot the graph of a linear function is to first make a table with a few $x$ values and the corresponding $y$ values and then plot each of these points. The points where the line intercepts the $x$- and $y$-axis are useful to make a note of.

Note: Another common way of writing linear functions is $y=mx+c$ where $m$ represents the slope and $c$ represents the $y$-axis intercept.

For more information about linear functions see Linear Functions and Equation of a Straight Line.

Gradient of a Line

The slope or gradient of a straight line is a measure of how steep the line is. It is the amount by which the value of the output $y$ changes when the value of the input $x$ increases by $1$. For instance, we might want to know by how much the quantity demanded changes when the price rises by $£1$.

We calculate the slope by dividing the change in the value of $y$ beween any two points on the line by the corresponding change in the value of $x$ between the same two points. This can be remembered as “rise over “run”: \[a=\dfrac{y_1 - y_0}{x_1 - x_0}=\frac{\triangle y}{\triangle x}\] Where $(x_0,y_0)$ and $(x_1,y_1)$ are two points on the function and $a$ is the slope of the function.

When the gradient is positive, the line goes upwards from left to right and increasing the gradient makes the line steeper. When the gradient is negative, the line goes downwards from left to right and decreasing the gradient makes the line steeper. A horizontal line has a gradient of zero.

For example, suppose you are running up a hill and that for every $20$ metres you move horizontally, you gain $5$ metres in height. The slope of the hill would be equal to \[\frac{\triangle y}{\triangle x}=\frac{5}{20}=0.25\text{ or } 25\%\]

Note: We can see by looking at the graph of any linear function that the slope of the line is constant: it is the same at every point along the line. Another way of thinking about this is that a given change in the value of $x$ always has the same effect on the value of $y$.

Intercepts

$Y$-axis Intercept

The $y$-axis intercept is the point where the graph of a function intersects with (“cuts”) the $y$-axis. In other words, it is the value that $y$ takes when $x=0$.

$X$-axis Intercept

The $x$-axis intercept is the point where the graph of a function intersects with the $x$-axis. It is the value that $x$ takes when $y=0$.

We can work out the $x$-axis intercept by rearranging the function to make $x$ the subject and setting $y$ equal to zero. For example, rearranging the function $y=2x+3$ to make $x$ the subject gives: \[x=\frac{y}{2}-\frac{3}{2}\] By setting $y$ equal to $0$ we can find the $x$-axis intercept: \begin{align} x&=\frac{0}{2}-\frac{3}{2}\\ &=\frac{3}{2} \end{align} So the $x$-axis intercept is $x=\frac{3}{2}$.

Note: All linear functions have only one $y$-axis intercept and one $x$-axis intercept due to the one-to-one mapping between $x$ and $y$: there can only be one value of $y$ corresponding to $x=0$ and one value of $x$ corresponding to $y=0$.

Worked Examples

Implicit Linear Functions

An explicit function is a function in which the dependent variable is written on the opposite side of the equals sign to the dependent variable. We say that the independent variable is written explicitly as a function of the independent variable. For example, the linear function: \[y=4x+9\] is an explicit function because $y$ is written as a function of $x$.

Sometimes we will be given an implicit function where both of the variables are on the same side of the equation. For example, we can rewrite the explicit function from above as an implicit function by rearranging it so that variables are on the same side of the equals sign: \[y-4x=9\] or \[y-4x-9=0\]

Note: Since rearranging a function does not change the properties of a function, the implicit and explicit forms of a function have the same properties. They are simply different ways of writing the same function.

External Resources

• Various resources on linear functions by mathcentre
• Equation of a straight line by Maths is Fun