is y=3x-2 a one to one function.?
yes. it passes the vertical line test AND the horizonal line test.
To determine if the function y = 3x - 2 is one-to-one, we need to check if every x-value corresponds to a unique y-value.
For a linear function like y = 3x - 2, a horizontal line test can be used to determine if it is one-to-one. The horizontal line test states that if a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one.
Let's first assume that the function y = 3x - 2 is not one-to-one. This would mean that there exist two different x-values that map to the same y-value.
Suppose there are two x-values, x₁ and x₂, such that y(x₁) = y(x₂). This would mean that:
3x₁ - 2 = 3x₂ - 2
If we rearrange the equation, we get:
3x₁ = 3x₂
Dividing both sides by 3, we have:
x₁ = x₂
This implies that the two assumed x-values, x₁ and x₂, are actually the same value. Therefore, our assumption that the function is not one-to-one is false.
Since there does not exist two different x-values that map to the same y-value, we can conclude that the function y = 3x - 2 is one-to-one.
To determine if the function y = 3x - 2 is one-to-one, we need to check if it passes the horizontal line test. The horizontal line test states that for a function to be one-to-one, any horizontal line must intersect the graph at most once.
To apply the horizontal line test to the given function, imagine drawing a horizontal line anywhere on the graph of y = 3x - 2. If no horizontal line intersects the graph at two or more points, then the function is one-to-one.
To visualize this, we can plot the points on a graph or use algebraic reasoning. Let's use algebraic reasoning:
Suppose there are two points on the graph (x₁, y₁) and (x₂, y₂) where x₁ ≠ x₂. If y₁ = y₂, then the function would not pass the horizontal line test and would not be one-to-one.
But, for any given x, the value of y can be uniquely determined using the equation y = 3x - 2. There is always exactly one output value for every input value.
Therefore, the function y = 3x - 2 is indeed one-to-one.