Is y=2x−1 a function?
The Correct Answer and Explanation is:
Yes, the equation ( y = 2x – 1 ) represents a function.
Explanation:
A function is a relationship between two sets, usually denoted as input (domain) and output (range), where each input value (usually ( x )) is paired with exactly one output value (usually ( y )). To determine if an equation represents a function, we can check if each input corresponds to only one output.
Vertical Line Test:
One way to determine if a relation is a function is by using the vertical line test. This test involves graphing the equation and checking if any vertical line drawn on the graph intersects the curve at more than one point. If a vertical line intersects the graph at more than one point, the equation is not a function. If it intersects at only one point, the equation is a function.
For the equation ( y = 2x – 1 ), it is a linear equation, which produces a straight line when graphed. A vertical line will intersect this straight line at exactly one point no matter where it is drawn, meaning the vertical line test confirms that ( y = 2x – 1 ) is indeed a function.
Mapping Input to Output:
In this equation, for every value of ( x ), there is exactly one corresponding value of ( y ). For example:
- If ( x = 1 ), then ( y = 2(1) – 1 = 1 ).
- If ( x = 0 ), then ( y = 2(0) – 1 = -1 ).
- If ( x = -1 ), then ( y = 2(-1) – 1 = -3 ).
Each input ( x ) gives only one output ( y ), which confirms that ( y = 2x – 1 ) is a function.
Since it meets the criteria that each input has one unique output, ( y = 2x – 1 ) is a valid function.