Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=3 $$

Step-by-Step Solution

Answer

The function \(f(x)=3\) is not one-to-one.

Step 1: Understand the Definition of a One-to-One Function

A function is considered one-to-one (injective) if and only if each value in the domain maps to a unique value in the codomain. In other words, if \(f(a) = f(b)\) implies \(a = b\), then the function is one-to-one.

Step 2: Analyze the Given Function

The function given is \(f(x) = 3\). This means no matter the input \(x\), the output is always 3. It is a constant function. Let's consider different values for \(x\).

Step 3: Check for Unique Mapping

Choose two different inputs, say \(x_1\) and \(x_2\) such that \(x_1 eq x_2\). For these inputs, \(f(x_1) = 3\) and \(f(x_2) = 3\). We see that different inputs yield the same output.

Step 4: Conclusion Based on Definition

Since for \(x_1 eq x_2\), \(f(x_1) = f(x_2)\) does not imply \(x_1 = x_2\), the function does not satisfy the condition for being one-to-one. Therefore, \(f(x) = 3\) is not a one-to-one function.

Key Concepts

Constant Function

A constant function is a simple yet essential concept in mathematics. It is a type of function where the output value remains the same regardless of the input. In the case of a constant function like \(f(x) = 3\), for any input \(x\), the output is always 3. This makes constant functions easy to spot on a graph—they are represented by a horizontal line across the specified range of \(x\) values.

Some key characteristics of constant functions include:

• Domain and Range: The domain of a constant function can be any set of real numbers. However, the range is a single value (in this case, 3).
• No Rate of Change: A unique aspect of constant functions is that they have no rate of change. The derivative of a constant function is always zero because the value does not change with respect to \(x\).
• Simplicity: They are among the simplest types of functions you can encounter in mathematics, making them an excellent starting point for understanding more complex function types.

Function Analysis

Function analysis involves examining different properties of a function to understand its behavior and characteristics. When analyzing functions, mathematicians often look at several aspects:

• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values the function can produce.
• Graphical Representation: Visualizing the function on a graph can offer insights into its behavior and important features such as intercepts and asymptotes.

Performing function analysis on \(f(x) = 3\) reveals:

• Domain: All real numbers.
• Range: Only the single value of 3.
• Graph: A horizontal line at \(y = 3\), showing no variation with different \(x\) values.

Function analysis helps identify whether functions have important properties, such as being constant or variable, continuous or discontinuous. It's a critical process that aids in the understanding and application of functions in various mathematical contexts.

Injective Function

An injective function, also known as a one-to-one function, is an important concept in mathematics that ensures each element of the domain maps to a distinct and unique element of the codomain. To determine if a function is injective, consider whether, for any two distinct elements \(a\) and \(b\) in the domain, \(f(a) = f(b)\) implies \(a = b\).

Key characteristics of an injective function include:

• Unique Mapping: Every input must have a unique output, ensuring no two different inputs are mapped to the same output.
• Graphical Representation: On a graph, any horizontal line should intersect the function at most once, confirming uniqueness of outputs.

In the case of \(f(x) = 3\):

• Since every input \(x\) maps to the output 3, different inputs do not yield unique outputs; thus, it's not injective.
• The horizontal line test does not apply, as the line would intersect the graph at multiple points, indicating multiple inputs map to the same output.

Understanding what makes a function injective is crucial when dealing with mappings in mathematics, as it ensures that each element in the domain has a distinct partner in the codomain, maintaining one-to-one correspondence.