To check if a function is one-to-one and onto, follow these steps:
Checking for One-to-One:
Step 1: Assume that \(f(a) = f(b)\), where \(a\) and \(b\) are two different values in the domain of the function.
Step 2: Simplify the equation and see if it leads to a contradiction or forces \(a = b\).
Step 3: If the equation leads to a contradiction, the function is one-to-one.
Step 4: If the equation forces \(a = b\), the function is not one-to-one.
Checking for Onto:
Step 1: Determine the range of the function by evaluating \(f(x)\) for different values of \(x\) in the domain.
Step 2: If every element in the range has a corresponding element in the domain, the function is onto.
Step 3: If there exist elements in the range that do not have a corresponding element in the domain, the function is not onto.
By following these steps, you can assess whether a function is one-to-one or onto.
To provide a visual understanding of one-to-one and onto functions, here are the graphical representations of each:
One-to-One Function Graph:
The horizontal line test is used to determine whether a function is one-one when its graph is given. to test whether the function is one-one from its graph,
For example, Consider the graph below.
Onto Function Graph:
To determine if a function is onto using its graph, a straightforward approach is to compare the range with the codomain. If the range is equal to the codomain, then the function is onto. For any function graph, it can be considered onto only if every horizontal line intersects the graph at one or more points. If there exists an element in the range that fails to intersect the function's graph when subjected to the horizontal line test, then the function is not onto. The following image exemplifies a graph of an onto function:
Here's a tabular format highlighting the key differences between one-to-one and onto functions:
Property
One-to-One Function
Onto Function
Each element in the domain maps to a unique element in the range.
Every element in the range has at least one corresponding element in the domain.
\(f(a) = f(b)\) implies \(a = b\), for all \(a\), \(b\) in the domain.
For every \(y\) in the range, there exists an \(x\) in the domain such that \(f(x) = y\).
Inputs and Outputs
No two different inputs produce the same output.
No elements in the range are left unmapped; every element has a corresponding input.
Passes the horizontal line test. The graph does not intersect any horizontal line at more than one point.
Covers the entire range vertically. The graph reaches every point in the range.
Injective or Surjective
Remember that a function can be both one-to-one and onto, which is referred to as a bijection.
1.Determine if the function \(f(x) = 2x - 3\) is one-to-one or onto.
Solution:
To check if \(f(x)\) is one-to-one, assume that \(f(a) = f(b)\), where \(a\) and \(b\) are two different values in the domain of \(f(x)\). Then we have:
Since \(a\) and \(b\) are equal, we can conclude that the function \(f(x)\) is one-to-one.
To check if \(f(x)\) is onto, we need to determine if every element in the range of \(f(x)\) has a corresponding element in the domain. The range of \(f(x)\) is all real numbers, since any real number can be obtained by plugging in a value for \(x\).
Therefore, f(x) is onto.
2.Determine if the function \(g(x) = x^ - 4x\) is one-to-one or onto.
Solution:
To check if \(g(x)\) is one-to-one, we assume that \(g(a) = g(b)\), where \(a\) and \(b\) are two different values in the domain of \(g(x)\). Then we have:
\(a^ - b^ - 4a + 4b = 0\)
\((a - b)(a + b) - 4(a - b) = 0\)
Since \((a - b)(a + b - 4) = 0\), either \(a - b = 0\) or \(a + b - 4 = 0\). If \(a - b = 0\), then \(a = b\), and we have proven that \(g(x)\) is one-to-one. However, if \(a + b - 4 = 0\), then we cannot conclude that \(g(x)\) is one-to-one.
To check if \(g(x)\) is onto, we need to determine if every element in the range of \(g(x)\) has a corresponding element in the domain. The range of \(g(x)\) is all real numbers greater than or equal to \(-4\). However, there is no real number that can be plugged in for \(x\) to obtain a negative number in the range of \(g(x)\).
Therefore, \(g(x)\) is not onto.
3.Determine if the function \(h(x) = 3x - 7\) is one-to-one or onto.
Solution:
To check if \(h(x)\) is one-to-one, assume that \(h(a) = h(b)\), where \(a\) and \(b\) are two different values in the domain of \(h(x)\). Then we have:
Since \(a\) and \(b\) are equal, we can conclude that the function \(h(x)\) is one-to-one.
To check if \(h(x)\) is onto, we need to determine if every element in the range of \(h(x)\) has a corresponding element in the domain. The range of \(h(x)\) is all real numbers, since any real number can be obtained by plugging in a value for \(x\).
Therefore, \(h(x)\) is onto.
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