# 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.

## Answer: Horizontal line testing and algebraic testing are two ways to determine a one-to-one function.

See detailed Explanation.

**Explanation:**

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) = x^{3 }+ 1 and g(x) = x^{2} - 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) = 5x^{3} - 1,

f(a) = f(b)

⇒ 5a^{3} - 1 = 5b^{3} - 1

⇒ 5a^{3 }= 5b^{3}

⇒ a^{3 }= b^{3}

⇒ ∛ a^{3 } = ∛ b^{3}

⇒ a = b

So, the function f(x) = 5x^{3} - 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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