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A function in mathematics is a relation with the property that every input is associated to exactly one output. Linear functions are extremely important. Furthermore, they appear in several equation forms. A linear function is one that has one or two variables but no exponents. On the coordinate plane, this function represents a straight line. If the function has extra variables, they must be constant in order to stay in the same linear function condition.
Definition:
A linear function is an algebraic function that produces a straight line in a coordinate plane. It is, in general, a polynomial function with a maximum degree of 1 or 0. Calculus and linear algebra are also used to express linear functions. The main distinction is in the function notation. An ordered pair stated in function notation is required.
A linear function has the formulaf(x) = mx + b, where m and b are real values.
A linear function is a function that represents the coordinate plane as a straight line. For instance, y =3x – 2 indicates a straight line on a coordinate plane and thus a linear function. This function can be expressed as f(x) = 3x – 2 since y can be replaced with f(x).
Where, The slope of the line is’m’ and the y-intercept is ‘b.’
The independent variable is ‘x.’
The dependent variable is ‘y’ (or f(x)
Linear Function Equations
The parent linear function is f(x) = x, which is a line running through the origin. a linear function equation is f(x) = mx + b, and here are some examples.
For example:
f(x) = 5x – 4
f(x) = -3x – 0.3
f(x) = 5
Linear functions graph examples
In Linear functions graph, Take any two points on a linear function and find the corresponding values of the other variable by assuming some random numbers for the dependent or independent variable. Simply draw a line connecting the 2 points.
To begin, locate the two points (x1,x2)and (y1,y2) that fulfil the equation y = mx+b.
Put these points on a graph or the X-Y axis.
To connect the two locations in the plane, draw a straight line.
Furthermore, the slope of a linear function is calculated as m=y2-y1x2-x1
Example: 1 Draw the following linear equation: 2 x + 2 y = 2
Solution: Assume that x = 0 and solve for y.
2 0+ y = 2
y = 2
Substitute y = 0 and solve for x.
2x + 0 = 2
2x = 2
x = 1
Draw a graph with the x and y intercept points (0, 2) and their intersections (1,0)
Connect the two locations using a straight line.
How to do linear functions step by step
A linear function is one that has the formula f(x) = ax+ b. It appears to be a standard linear equation, however instead of y, the linear function notation is f. (x). You would be given the value of f(x) and asked to find x to solve a linear function.
Step for how to solve linear function given below:
Fill in the value of f(x) in the issue. In this instance: 3x – 1 = 8
Remove the variable. In this situation, you add 1 to both sides to isolate the variable term before moving the constant term over the equal sign using the reverse operation. As a result
8 + 1 = 3x – 1 + 1
9 = 3x
Isolate the variable further. In this situation, divide both sides by 3 to isolate the variable itself. As a result:
93 = 3×3
Simplify. So we have
x = 3
Because this is a linear function, the final answer should be in function form.
f(x) = 3x – 1, f(3) = 8.
Linear Function properties
A variable is a symbol in an expression that represents a quantity.
The steepness of a linear function represents the rate at which it deviates from a reference.
Linear functions can be growing, decreasing, horizontal, or vertical in direction.
A decreasing linear function has a slope that is negative. As a result, if m0, then f(x) = mx + b decreases.
A rising linear function is one with a positive slope. As a result, when m>0, f(x)=mx+b increases.
The y-intercept is the value of a function when the input value is zero. It is referred to as the beginning value.
Conclusion
We learned in this article that the term linear function refers to two distinct but related concepts: A linear function in calculus and related fields is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. In economics, the linear function is often used. It is appealing because it is simple and straightforward mathematically. It has a wide range of applications. Linear functions have a straight line as their graph.
