Determining whether a function is one-to-one is a crucial step in understanding its behavior and properties. A function is said to be one-to-one if each element in its domain is mapped to a unique element in its range. In other words, no two elements in the domain can map to the same element in the range. In this article, we will discuss how to determine if a function is one-to-one.
The Vertical Line Test
One of the easiest ways to determine if a function is one-to-one is to use the vertical line test. To perform the vertical line test, draw a vertical line through the graph of the function. If the vertical line intersects the graph at more than one point, the function is not one-to-one. However, if the vertical line intersects the graph at only one point, the function is one-to-one.
The Horizontal Line Test
Another way to determine if a function is one-to-one is to use the horizontal line test. To perform the horizontal line test, draw a horizontal line through the graph of the function. If the horizontal line intersects the graph at more than one point, the function is not one-to-one. However, if the horizontal line intersects the graph at only one point, the function is one-to-one.
The Algebraic Test
We can also determine if a function is one-to-one algebraically by using the definition of one-to-one functions. Recall that a function is one-to-one if each element in its domain is mapped to a unique element in its range. Therefore, we can use the following test to determine if a function is one-to-one:
- Let f(x1) = f(x2) for some x1 and x2 in the domain of f.
- If x1 = x2, then f(x1) = f(x2) implies x1 = x2.
- If x1 Γëá x2, then f(x1) = f(x2) implies that the function is not one-to-one.
In other words, if we can find two distinct elements in the domain of the function that map to the same element in the range, the function is not one-to-one.
Examples
Let’s consider some examples to illustrate the different tests for one-to-one functions.
Example 1: f(x) = x^2
To determine if f(x) = x^2 is one-to-one, we can use the vertical line test. If we draw a vertical line through the graph of f(x) = x^2, we can see that it intersects the graph at more than one point, as shown below:
Therefore, f(x) = x^2 is not one-to-one.
Example 2: f(x) = 2x + 1
To determine if f(x) = 2x + 1 is one-to-one, we can use the algebraic test. Suppose f(x1) = f(x2) for some x1 and x2 in the domain of f. Then, we have:
2×1 + 1 = 2×2 + 1
2×1 = 2×2
x1 = x2
Therefore, f(x) = 2x + 1 is one-to-one.
Example 3: f(x) = sin(x)
To determine if f(x) = sin(x) is one-to-one, we can use the horizontal line test. If we draw a horizontal line through the graph of f(x) = sin(x), we can see that it intersects the graph at more than one point, as shown
