A linear function is a function that represents a straight line on the coordinate plane. For example, y = 3x  2 represents a straight line on a coordinate plane and hence it represents a linear function. Since y can be replaced with f(x), this function can be written as f(x) = 3x  2.
A linear function is of the form f(x) = mx + b where 'm' and 'b' are real numbers. Isn't it looking like the slopeintercept form of a line which is expressed as y = mx + b? Yes, this is because a linear function represents a line, i.e., its graph is a line. Here,
A linear function is an algebraic function. This is because it involves only algebraic operations.
The parent linear function is f(x) = x, which is a line passing through the origin. In general, a linear function equation is f(x) = mx + b and here are some examples.
Here are some reallife applications of the linear function.
We use the slopeintercept form or the pointslope form to find a linear function. The process of finding a linear function is the same as the process of finding the equation of a line and is explained with an example.
Example: Find the linear function that has two points (1, 15) and (2, 27) on it.
Solution: The given points are (x_{1}, y_{1}) = (1, 15) and (x_{2}, y_{2}) = (2, 27).
Step 1: Find the slope of the function using the slope formula:
m = (y_{2}  y_{1}) / (x_{2}  x_{1}) = (27  15) / (2  (1)) = 12/3 = 4.
Step 2: Find the equation of linear function using the point slope form.
y  y_{1} = m (x  x1)
y  15 = 4 (x  (1))
y  15 = 4 (x + 1)
y  15 = 4x + 4
y = 4x + 19
Therefore, the equation of the linear function is, f(x) = 4x + 19.
If the information about a function is given as a graph, then it is linear if the graph is a line. If the information about the function is given in the algebraic form, then it is linear if it is of the form f(x) = mx + b. But to see whether the given data in a table format represents a linear function:
Example: Determine whether the following data from the following table represents a linear function.
Solution: We will compute the differences in xvalues, differences in y values, and the ratio (difference in y)/(difference in x) every time and see whether this ratio is a constant.
Since all numbers in the last column are equal to a constant, the data in the given table represents a linear function.
We know that to graph a line, we just need any two points on it. If we find two points, then we can just join them by a line and extend it on both sides. The graph of a linear function f(x) = mx + b is
There are two ways to graph a linear function.
To find any two points on a linear function (line) f(x) = mx + b, we just assume some random values for 'x' and substitute these values in the function to find the corresponding values for y. The process is explained with an example where we are going to graph the function f(x) = 3x + 5.
To graph a linear function, f(x) = mx + b, we can use its slope 'm' and the yintercept 'b'. The process is explained again by graphing the same linear function f(x) = 3x + 5. Its slope is, m = 3 and its yintercept is (0, b) = (0, 5).
Graphing a Linear Function Using SlopeIntercept Form
The domain of a linear function is the set of all real numbers, and the range of a linear function is also the set of all real numbers. The following figure shows f(x) = 2x + 3 and g(x) = 4 −x plotted on the same axes.
Note that both functions take on real values for all values of x, which means that the domain of each function is the set of all real numbers (R). Look along the xaxis to confirm this. For every value of x, we have a point on the graph.
Also, the output for each function ranges continuously from negative infinity to positive infinity, which means that the range of either function is also R. This can be confirmed by looking along the yaxis, which clearly shows that there is a point on each graph for every yvalue. Thus, when the slope m ≠ 0,
Note: (i) The domain and range of a linear function is R as long as the problem has not mentioned any specific domain or range.
(ii) When the slope, m = 0, then the linear function f(x) = b is a horizontal line and in this case, the domain = R and the range = {b}.
The inverse of a linear function f(x) = ax + b is represented by a function f^{1}(x) such that f(f^{1}(x)) = f^{1}(f(x)) = x. The process to find the inverse of a linear function is explained through an example where we are going to find the inverse of a function f(x) = 3x + 5.
Note that f(x) and f^{1}(x) are always symmetric with respect to the line y = x. Let us plot the linear function f(x) = 3x + 5 and its inverse f^{1}(x) = (x  5)/3 and see whether they are symmetric about y = x. Also, when (x, y) lies on f(x), then (y, x) lies on f^{1}(x). For example, in the following graph, (1, 2) lies on f(x) whereas (2, 1) lies on f^{1}(x).
Sometimes the linear function may not be defined uniformly throughout its domain. It may be defined in two or more ways as its domain is split into two or more parts. In such cases, it is called a piecewise linear function. Here is an example.
Example: Plot the graph of the following piecewise linear function.
Solution: This piecewise function is linear in both the indicated parts of its domain. Let us find the endpoints of the line in each case.
When x ∈ [2, 1):
When x ∈ [1, 2]:
The corresponding graph is shown below:
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