What form of correspondence does the function f(x) = 2x + 5 have?
The Correct Answer and Explanation is :
The function ( f(x) = 2x + 5 ) is an linear function.
Explanation:
A linear function is a type of function where the highest degree of the variable ( x ) is 1. The general form of a linear function is given by:
[
f(x) = mx + b
]
where:
- ( m ) is the slope of the line, and
- ( b ) is the y-intercept, or the point where the line crosses the y-axis.
In the function ( f(x) = 2x + 5 ):
- The slope ( m ) is 2. This means for every 1 unit increase in ( x ), ( f(x) ) (or ( y )) increases by 2 units.
- The y-intercept ( b ) is 5, indicating that the line crosses the y-axis at ( y = 5 ) (when ( x = 0 )).
Graphical Representation:
The graph of a linear function is always a straight line. For ( f(x) = 2x + 5 ), the line will have a slope of 2 and will pass through the point ( (0, 5) ) on the y-axis. The line extends infinitely in both directions, maintaining a constant slope.
Properties of Linear Functions:
- Constant rate of change: The slope ( m ) represents the constant rate of change between ( x ) and ( f(x) ). In this case, for every unit increase in ( x ), the value of ( f(x) ) increases by 2 units.
- Straight line: Since the function is linear, its graph is a straight line, with no curves or bends.
- No exponents higher than 1: The variable ( x ) only has an exponent of 1, which is characteristic of a linear function. Functions with exponents greater than 1 are called polynomial functions of higher degree (quadratic, cubic, etc.).
Thus, ( f(x) = 2x + 5 ) is a linear function with a slope of 2 and a y-intercept of 5.