How to Unravel One-to-One Functions

Step 1: Defining One-to-One Functions

• The Basic Idea: At its core, a function is one-to-one if no two different inputs have the same output.
• Mathematically Speaking: A function \(f\) is one-to-one if, for every pair of different inputs \(a\) and \(b\), their outputs \(f(a)\) and \(f(b)\) are also different. If \(f(a)=f(b)\), then \(a\) must equal \(b\).

Step 2: Visualizing with Graphs

• Horizontal Line Test: The quickest way to determine if a function is one-to-one graphically is by using this test. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one.
• Understanding the Test: The reason for this is simple: a horizontal line represents a constant output. If it touches the function at two points, then two distinct inputs share the same output, violating the definition of a one-to-one function.

Step 3: Investigating Algebraically

• Expressing \(x\) in terms of \(f(x)\): If you can solve an equation for \(x\) and get a unique solution for \(x\) in terms of \(f(x)\), it’s a strong indication that the function is one-to-one.
• Unique Solutions Matter: The solution for \(x\) must be unique. Multiple solutions would mean multiple inputs for a single output, which isn’t allowed for one-to-one functions.

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Step 4: Exploring the World of Inverses:

• What’s an Inverse?: Given a function \(f\), its inverse, represented as \(f^{−1}\), switches the roles of inputs and outputs. Essentially, it “undoes” the function.
• Existence of Inverse: Not all functions have inverses that are also functions. However, every one-to-one function has a unique inverse that is itself a function.
• Graphical Reflection: On a graph, the function and its inverse are symmetrical about the line \(y=x\). This is a visual representation of their interconnected roles.
• Functional Dance: A unique property of functions and their inverses is that \(f(f^{−1}(x))=x\) and \(f^{−1}(f(x))=x\). This shows that they perfectly counteract each other.

Step 5: Appreciating the Importance

• In Computing, One-to-one functions, especially in the realm of algorithms, ensure that data remains distinct after processing.
• In Science, Predictability is crucial in experiments. One-to-one relations guarantee that a unique set of conditions or states yields a distinct result.

Final Words

Examples:

Example 1:

Consider the function \(g(x)=3x−4\). Is this a one-to-one function?

Solution:

Assume \(g(a)=g(b)\), where \(a\) and \(b\) are elements in the domain.

Starting with:

\(3a−4=3b−4\)

Add 4 to both sides:

\(3a=3b\)

Now, divide both sides by \(3\):

\(a=b\)

Since our assumption leads directly to \(a=b\), the function \(g(x)\) is one-to-one.

Example 2:

Determine whether the function \(h(x)=x^3\) is a one-to-one function.

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Solution:

Let’s assume \(h(a)=h(b)\), where \(a\) and \(b\) are elements in the domain.

This leads us to:

\(a^3=b^3\)

Taking the cube root of both sides gives:

\(a=b\)

In this case, our assumption directly leads to \(a=b\). Therefore, unlike \(x^2\), the function \(h(x)=x^3\) is one-to-one.

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