One-to-one Functions​

One-to-one Functions​

Example No. 1

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​The relation pairing an SSS member to his or her SSS number.

Example No. 1

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​The relation pairing an SSS member to his or her SSS number.

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​ONE-TO-ONE FUNCTION

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​Each SSS member is assigned a unique SSS number. Thus, this relation is a

function. Further, two members cannot be assigned the same SSS number,

therefore, the function is one-to-one.

Example No. 2

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The relation pairing a citizenship to a person.

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Example No. 2

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The relation pairing a citizenship to a person.

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​NOT A ONE-TO-ONE FUNCTION

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The relation is a function because each person has a citizenship. However, a person can have two citizenship, (dual citizen) therefore, it is not one-to-one function.​

​Graph of a One-to-one Function

​If f is a one-to-one function then no two points (x₁, x₂) and (y₁, y₂) have the same y-value. Therefore, no horizontal line cuts the graph of the equation y = f(x) more than once.

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​The Inverse of One-to-one Function

​Exercise:

1. ​Think of a number.

2. ​Multiply it by 2.

3. ​Then, subtract 1 from it.

4. ​Now, add 4 to the difference.

5. ​Lastly, give me your answer and I’ll tell the number you are thinking of.

Inverse Function​

The inverse of a function is a function with domain B and range A given that the original function has domain A and range B.

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​This inverse function of function f is denoted by f^-1. It is defined by the equation 𝑓⁻¹(𝑦) = 𝑥, if and only if, 𝑓(𝑥) = 𝑦 for any y in range B.

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Since both are functions, then a function has to be one-to-one for its inverse to be a function at the same time. If it is a many-to-one function, its inverse is one-to-many which is not a function.

How to find the inverse of a one-to-one function?

​Example 1: ​

𝑓(𝑥) = 3𝑥 – 8​

STEP1: The last operation performed is subtraction, the inverse

operation of which is addition. To x, add 8.

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​STEP2: The second to the last operation performed is multiplication, the inverse operation of which is division. Divide x + 8 by 3.

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​STEP3: Equate it to 𝑓−1(𝑥) to denote that it is the inverse function of 𝑓(𝑥) = 3𝑥 – 8.

To find the inverse of a one-to-one function, consider the following:​

• Express the function in the form 𝑦 = 𝑓(𝑥);

• Interchange the x and y variables in the equation;

• Solve for y in terms of x.​

​Example 2: ​

𝑔(𝑥) = 𝑥² – 6𝑥 – 7

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