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Linear Functions

Mathematics is full of functions and relations. A relation is the collection of well defined ordered pairs. A function is a relation to a property that corresponding to every permitted input there exists one single output. A polynomial function is a very useful function. The degree of a polynomial function is defined as the highest variable exponent in the function. A polynomial function of single degree is defined as a linear function, i.e. if the highest power of the variable in the polynomial function is one, then it is known as a linear function. A mathematical equation in which there is no independent-variable is raised to a power greater than one.

The graph of a linear function consists of the segments of one straight line throughout its domain, i.e. a linear function represents a straight line. In general, a linear function is expressed as -

y = m x + c
or
f(x) = m x + c

where, m is the slope or gradient of the line and c is its intercept on Y-axis. It traces a straight line when plotted on a graph. It is also called as a linear equation. In this page below, we are going to focus on linear function, its definition, its properties and various problems based on linear functions.

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How to Write a Linear Function?

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The function is defined by f the first degree equation: f = { ( X, Y)/ Y = m X + b } where, m and b are constants, x and y is called a linear functions. The function derives a straight line while graphing. A simple linear function with one independent variable can be written as y = a + bx, where a and b real constants. The constant b is the proportional change of y as x varies. The constant a is the value of y when the value of x is 0.

Functions such as these gives graph that are straight lines, and, thus, the name linear. There are three main forms in linear functions. They are as follows,

  1. Slope-Intercept Form is given by y = mx +b.
  2. Point Slope Form is given by m = $\frac{y - y_1}{x - x_1}$.
  3. General Form is given by Ax + By + C = 0

Derivative of a Linear Function

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The derivative of any linear function f(x) = y = mx + c, is finding the differentiating the y with respect to x. Since we have a linear relation in which x and y are  linear form hence if we differentiate y with respect to x we get only m which is called the slope of the tangent of ant linear equation.

Solved Example

Question: Find y' if y = f(x) = 4x + 3
Solution:
For y', we can differentiate y with respect to x, then
$\frac{dy}{dx}$ = $\frac{d}{dx}$ $f(x)$

= $\frac{d}{dx}$ $(4x + 3)$

= $4$


Inverse of a Linear Function

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To find the inverse of a linear function is quite easy as compared to the inverse of any rational or quadratic function, since the domain  and range of any linear function is all real numbers. For finding the inverse of any linear function first we have to interchange x and y in the equation and after that solve that equation for y.

Solved Example

Question: Find the inverse of the function $f(x)$ = $\frac{6x + 4}{5}$
Solution:
For finding the inverse of f(x), set the equation as

$y$ = $\frac{6x + 4}{5}$

Switch x and y, then

$x$ = $\frac{6y + 4}{5}$

$5x = 6y + 4$

$y$ = $\frac{5x - 4}{6}$

$y = f^{-1}(x)$ = $\frac{5x - 4}{6}$

The domain of the linear function f(x) is the set all possible real values, which allow the function formula  to work. The range of any linear function is the set of all out put values which can be found after putting the input values of that function.

Solved Example

Question: Find the domain and range of the function y = 4x +1.
Solution:
Let f(x) = y = 4x +1. then for finding the domain and range of the function, let
x = 0 then y = 1
x = 1 then y = 5
x = 2 then y = 9
x = 3 then y = 13
x = -1 then y = -3 etc.
So, we get the set of ordered pairs as (0,1), (1,5), (2, 9), (3, 13) , (-1, -3) etc.
So, the domain of the given function is (-1, 0, 1, 2, 3...etc.)

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Linear Function Table

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Sometimes the information of any function is written in the tabulated form. If the input values in the data table are in the form of evenly spaced then the function is linear function and the out put values are in the form of evenly spaced. But if the input values in the data table are in the form of evenly spaced and if the output values not in the form of evenly spaced then the function is not in linear form.

sometimes the input values in the table not in the form of evenly spaced then we have to find the rate of change between the all successive pairs which are presents in the table. In the graph of all line segments of these points have the same slope then the function is known as the linear function.
If we have a tabular data as follows, then the rate of change shows that the function is a linear function.
Linear Function Table

Piecewise Linear Function

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A function defined over an equal number of intervals and formed by some number of linear line segments is called the piecewise linear function. they gives the information about the set of slopes, breaking point at which the slope changes and the value of the function at given points.

For a piecewise linear function the length of a piece is given by

$\sqrt{(\bigtriangleup x)^{2}+ (\bigtriangleup y)^{2}} = \sqrt{1 + \left ( \frac{\bigtriangleup y}{\bigtriangleup x}^{2} \right )}\bigtriangleup x$

If we summing up the length of all of the pieces, we get
$\sum_{j = 1}^{n}\left [ \sqrt{1+\left ( \frac{\bigtriangleup y_{j}}{\bigtriangleup x_{j}} \right )^{2}} \bigtriangleup x_{j}\right ]$

Taking the limit $\bigtriangleup x_{j}\rightarrow 0$, then we have

$\int \sqrt{1 + \left ( \frac{dy}{dx} \right )^{2}}dx$

Find a Linear Function

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Solving the given function by altering its position and simplifying the equation to get Y is generally known as solving linear equations. The solutions of these equations provide solutions of the corresponding practical problems.

The important steps in the solution of these word problems are listed below:

  1. Read the problem carefully and note down (i) what is given, and (ii) what is required.
  2. Denote the unknown quantity by a literal x, y, z, u, v, w etc.
  3. Translate the statements of the problem step by step into mathematical statements, to the extent possible.
  4. Look for the quantities that are equal. Write the equations corresponding to these equality relations.
  5. Solve the equations written in Step 4 above.
  6. Check the solution by substituting the value of the unknown found in Step 5 above into the statements of the problem.

Linear Function Equation

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A relation y = f(x) is called a function if for every value of x there is a corresponding value of y.y, where x is an independent variable and y a dependent variable.

Linear Equations: Linear Equations are of the form y = mx + b, where m is the slope and b- the y-intercept. In a linear equation, the linear relationship is such that when the independent quantity increases the dependent quantity increases or decreases, such that the ratio of change in y over the change in x is always constant.

Slope formula: The above constant ratio is called the slope of the line joining the points containing the independent and dependent variables in the form (x,y).

The slope of the line joining any two points ( x1, y1) and (x2, y2 ) is m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

The slope , y-intercept form of a linear equation is y = mx + b, where m is the slope of the line and b the y-intercept. The graph of a linear function is always straight line.

Transforming Linear Functions: The standard form of a linear function is f(x,y) = ax + by + c. The corresponding linear equation is ax + by + c=0.

Transformation of linear equation means, transforming the given equation into slope intercept form.
Consider the example, 10x + 5y = 3
5y = -10x + 3
y = -(10/5)x + (3/5) = -2x + 0.6
which is of the form y = mx + b, where the slope is m = -2 and the y-intercept b = 0.6.

Graphing the above linear function:

Step 1: Plot the points on the graph sheet.
Step 2: By joining the points we get the straight line which is the graph of the given linear function.

Linear Function Equation

The graph shows the points (0,2), (2,5), 4,8), 6,11), (8,14)

Linear Function Equations

The graph shows the straight line drawn by joining the points. As the value of x increases, the value of y also increases. From the above graph, we can easily predict the population in the given year and find the year during for which the population is given.

In a linear function, when the value of one variable increases, the values of other variable increases or decreases. In the above graph, as x increases y also increases.

Linear Function

Linear Functions

In the adjacent graph, we observe that as the values of x increases, y decreases.

Linear Function Graph

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To graph a linear function we perform the following steps.
  1. Equating the function f(x,y) = 0, we get a linear equation. 
  2. Find the x-intercept by plugging in y = 0.
  3. Find the y-intercept by plugging in x = 0.
  4. Plot the x and y-intercepts on the two-dimensional plane, and join the points and extend on both the directions.

Solved Example

Question: Graph the linear function f(x,y) = 4x + 5y - 20
Solution:
Step 1: The linear equation is 4x + 5y -20=0
4x + 5y = 20

Step 2: To find the x-intercept, plug in y = 0

4x + 5(0) = 20
4x + 0 = 20
4x = 20
x = $\frac{20}{4}$ = 5

The x-intercept is (5,0)

Step 3: To find the y-intercept, plug in x = 0.

4(0) + 5y = 20
0 + 5y = 20
5y = 20
y = $\frac{20}{5}$ = 4

The y-intercept is (0, 4)

Step 4: Plot the points (5,0) and (0,4) in the graph.

Linear Function Graph

Linear Functions Examples

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Given below are some of the examples on linear functions.

Solved Examples

Question 1: Solve for x and y, where y = x + 13 and 2x - y - 10 = 0
Solution:
Step 1: Plug y = x + 13 in 2x - y - 10 = 0

2x - (x + 13) - 10 = 0

x - 23 = 0

x = 23

Step 2: Plug x = 23 in 2x - y - 10 = 0

46 - y - 10 = 0

- y + 36 = 0

(- y) = (- 36)

y = 36

Step 3: x = 23 and y = 36



Question 2: Solve for x and y, where y = 2x + 3 and x + 3 = y
Solution:
Step 1: Given y = 2x + 3 and x + 3 = y

Step 2: Put x = y - 3 in y = 2x + 3
y = 2(y - 3) + 3
y = 2y - 2.3 + 3
y = 2y - 6 + 3
y = 2y - 3
2y - y = 3
y = 3

Step 3: Put y = 3 in x + 3 = y
x + 3 = 3
x = 3 - 3
x = 0
Step 4: Value of x = 0 and y = 3



Question 3: Find the value of x and y from linear equations x - y = 0 and 7x - 2 = y
Solution:
Step 1: Given x - y = 0, 7x - 2 = y

Step 2: Put y = x in 7x - 2 = y
7x - 2 = x
7x - x - 2 = 0
6x - 2 = 0
6x = 2
x = 2 / 6 = 1/3

Step 3: x = 1 / 3 put in x - y = 0
1 / 3 - y = 0
y = 1 / 3

Step 4: x = 1/3 and y = 1/3.



Question 4: Find the value of x and y from linear equations 2x + y = 0 and x - 2 = y
Solution:
Step 1: Given 2x + y = 0, x - 2 = y

Step 2: Put y = - 2x in x - 2 = y
x - 2 = - 2x
x + 2x - 2 = 0
3x - 2 = 0
3x = 2
x = 2 / 3

Step 3: x = 2 / 3 put in 2x + y = 0
2 . 2/3 + y = 0
y = - 4 / 3

Step 4: x = 2/3 and y = - 4/3



Linear Function Word Problems

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Given below are some of the word problems on linear function.

Solved Examples

Question 1: The taxi fare in a city is as follows. For the first 10 kilometer the fare is 50 dollars and for the subsequent km the fare is 4 dollars. Taking the distance covered is x km[ where x > 10 ] and total fare is y dollars, write the linear equation for the information.
Solution:
For the first 10 km, 50 dollars
For 11 km, fare = 50 + (1)4 = 50 + (11 - 10)(4)
For 12 km, fare = 50 +(2)4 = 50 + (12 - 10)(4)
For 13 km, fare = 50 + (3)4 = 50 + (13 - 10)(4)
For x km, fare = 50 + (x - 10)(4)
= 50 + 4x – 40 [ applying distributive property ]
= 4x + 10
Hence, the linear equation to calculate the fare is y = 4x + 10, where x ≥ 10.

Question 2: Find any two solutions of the linear equation, 2x + 3y = 6
Solution:
As the graph of the linear equation is a straight line, the domain consists of all real numbers. Hence, there are infinite solutions to the line. We can find different solutions by plugging in different values for x.
Let x = 1, 2(0) + 3y = 6
  • 0 + 3y = 6
  • y = 6/3 = 2
  • One of the solution is (0,2)
Let x = -3, then 2x + 3(-3) = 6
  • 2x - 9 = 6
  • 2x = 15
  • x = 15/3 = 5
  • The other solution is (-3,5)
Hence, the two solutions are (0,2) and (-3,5)

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