Linear functions are some of the most basic functions in mathematics yet extremely important to understand because they are widely applied in electronics, physics, economics, chemistry, and more. Also, several concepts in the theory of functions and related topics depend strongly on the concept of linear functions.
Definition and Properties of the Linear Functions
A linear function has the form:
where \( f \) is the name of the function, \( x \) is the variable, and \( a \) and \( b \) are constants such that \( a \neq 0 \).
The linear function as defined above gives an output for any value of the variable \( x \) in the set of real numbers. Therefore, the domain of any linear function is the set of all real numbers unless it is defined otherwise.
The graph of a linear function is a line with y-intercept at the point \( (0, b) \) and slope \( a \). If we take any two points \( P_1 \) and \( P_2 \) on the graph of the linear function \( f \), the slope \( a \) is given by:
- If \( a > 0 \), the line rises from left to right, and we say that \( f \) is an increasing function as \( x \) increases.
- If \( a < 0 \), the line falls from left to right, and we say that \( f \) is a decreasing function as \( x \) increases.
- The range of a linear function with \( a \neq 0 \) is the set of all real numbers.
The domain and range of a linear function are written in interval forms as follows:
Domain: \( (-\infty, +\infty) \)
Range: \( (-\infty, +\infty) \)
Examples and Detailed Solutions
Example 1: Graphing Linear Functions
a) Graph the linear functions given by \( f(x) = x + 3 \) and \( g(x) = 0.5 x + 3 \) on the same system of coordinates.
b) Which of the two functions increases faster?
View Solution
a) The graph of a linear function is a line, and two points only are needed to graph it. Let us find the values of the functions at \( x = 0 \) and \( x = 2 \):
| \( x \) | \( 0 \) | \( 2 \) |
|---|---|---|
| \( y = f(x) = x + 3 \) | \( 0 + 3 = 3 \) | \( 2 + 3 = 5 \) |
| \( x \) | \( 0 \) | \( 2 \) |
|---|---|---|
| \( y = g(x) = 0.5 x + 3 \) | \( 0.5(0) + 3 = 3 \) | \( 0.5(2) + 3 = 4 \) |
Two points for \( f \): \( (0, 3) \) and \( (2, 5) \).
Two points for \( g \): \( (0, 3) \) and \( (2, 4) \).
b) From the graph, we conclude that function \( f \) increases faster than function \( g \). In general, given two linear functions \( f \) and \( g \) with slopes \( m_1 \) and \( m_2 \):
- If both \( m_1 \) and \( m_2 \) are positive and \( m_1 > m_2 \), \( f \) increases faster than \( g \).
- If both \( m_1 \) and \( m_2 \) are negative and \( m_1 < m_2 \), \( f \) decreases faster than \( g \).
- If \( m_1 \) and \( m_2 \) have different signs, the one with positive slope increases and the one with negative slope decreases.
Example 2: Finding Linear Functions
Find the linear function \( f \) such that \( f(-1) = 4 \) and \( f(2) = 1 \).
View Solution
Being a linear function, \( f \) is of the form \( f(x) = ax + b \). We need to find constants \( a \) and \( b \):
\( f(-1) = 4 \implies -a + b = 4 \)
\( f(2) = 1 \implies 2a + b = 1 \)
We solve the system of equations:
\[ \begin{cases} -a + b = 4 \\ 2a + b = 1 \end{cases} \]Subtracting the first equation from the second to eliminate \( b \):
\[ (2a + b) - (-a + b) = 1 - 4 \implies 3a = -3 \implies a = -1 \]Substitute \( a = -1 \) into the first equation:
\[ -(-1) + b = 4 \implies 1 + b = 4 \implies b = 3 \]The function is given by:
\[ f(x) = -x + 3 \]Examples of Applications of Linear Functions
- Electronics: The voltage \( V \) across a resistor of resistance \( R \) is given by \( V = R I \), where \( I \) is the current.
- Physics: A resultant force \( F \) acting on an object of mass \( M \) is given by \( F = M a \), where \( a \) is acceleration.
- Chemistry: To convert degrees Celsius \( C \) into Kelvin \( K \), we use \( K = C + 273.15 \).
- Motion: The distance \( d \) covered by an object moving at average speed \( s \) during time \( t \) is \( d = s t \).
- Economics: The total cost \( C \) of \( x \) units is given by \( C = ax + C_0 \), where \( a \) is cost per unit and \( C_0 \) is fixed cost.
Interactive Tutorial to Further Explore Linear Functions
The properties of the graphs of linear functions are explored interactively using an app. The exploration is carried out by changing the parameters \( a \) and \( b \) included in the linear function \( f(x) = ax + b \).
Tutorial Questions and Answers
- Set parameter \( a \) to \( 1 \) and change parameter \( b \). How does the y-intercept change as \( b \) changes? Give a quantitative answer and explain it analytically.
- Set parameter \( b \) to any value and change parameter \( a \). For what values of parameter \( a \) is the graph of function \( f \) increasing? Decreasing? Constant?
- The graph of \( f \) is a line. Set \( a \) to a value and use two points on the graph to find the slope of the line. Compare the value of the slope found to the value of parameter \( a \). What does \( a \) represent?
- What is the domain of the linear function \( f \)?
- What is the range of function \( f \) when parameter \( a \neq 0 \)? What is the range when \( a = 0 \)?
Answer
If we set \( x = 0 \) in \( f(x) = ax + b \), we obtain \( f(0) = b \). The y-intercept of the graph of \( f \) is the point with coordinates \( (0, b) \).
Answer
- If \( a \) is positive, \( f \) is an increasing function on \( (-\infty, +\infty) \).
- If \( a \) is negative, \( f \) is a decreasing function on \( (-\infty, +\infty) \).
- If \( a \) is equal to \( 0 \), \( f \) is a constant function on \( (-\infty, +\infty) \).
Answer
The graph of function \( f \) is a line, hence the name linear function. Parameter \( a \) represents the slope of this line.
Answer
The domain of all linear functions is the set of all real numbers represented by the interval \( (-\infty, +\infty) \).
Answer
- If \( a \neq 0 \), the range of any linear function is the set of all real numbers represented by \( (-\infty, +\infty) \).
- If \( a = 0 \), \( f(x) = b \) is a constant function, and its range is the set \( \{b\} \).