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How do you determine if a function is one-to-one? | Mathematics for Digital SAT PDF Download
How do you determine if a function is one-to-one?
If the set of inputs have a single output every time, then it is said to be a function. Every function has a domain and a range.
Horizontal line testing and algebraic testing are two ways to determine a one-to-one function.
- A domain is the set of inputs or possible values of x for a function f(x), to make the equation true.
- A range is the set of outputs/solutions for the given output for the function f(x) or y.
- One To One functions are the functions that have a unique value of 'x' for every 'y'.
- For example: Let the function (x + 3) be the a one-to-one function. Therefore f(x) = y
To determine that whether the function f(x) is a One to One function or not, we have two tests.
(1) Horizontal Line testing: If the graph of f(x) passes through a unique value of y every time, then the function is said to be one to one function.
For example Let f(x) = x3 + 1 and g(x) = x2 - 1
- In the above graphs, the function f (x) has only one value for y and is unique, whereas the function g(x) doesn't have one-to-one correspondence.
- For function g(x), the graph is passing through two ordered pairs. So, it doesn't have a unique y for every x. Therefore, g(x) is not a one-to-one function.
(2) Algebraic Testing: The function is said to be one to one if a = b for every f(a) = f(b)
Let, f(x) = 5x3 - 1,
f(a) = f(b)
⇒ 5a3 - 1 = 5b3 - 1
⇒ 5a3 = 5b3
⇒ a3 = b3
⇒ ∛ a3 = ∛ b3
⇒ a = b
So, the function f(x) = 5x3 - 1 is one-to-one.
Thus, horizontal line testing and algebraic testing are two ways to determine a one to one function.
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