Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$f(x)=x^{3}$$

Step-by-Step Solution

Answer

The function \( f(x) = x^3 \) is one-to-one.

Step 1: Understanding One-to-One Function

A function is one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable. In mathematical terms, a function \( f(x) \) is one-to-one if \( f(a) = f(b) \) implies \( a = b \).

Step 2: Setting the Equation

To determine if the function \( f(x) = x^3 \) is one-to-one, set \( f(a) \) equal to \( f(b) \):\[ a^3 = b^3 \]

Step 3: Solving the Equation

From \( a^3 = b^3 \), take the cube root of both sides to solve for \( a \) and \( b \):\[ a = b \]

Step 4: Conclusion

Since \( a = b \) whenever \( a^3 = b^3 \), the function \( f(x) = x^3 \) satisfies the condition for being one-to-one. No two different input values yield the same output.

Key Concepts

Function Analysis

Function analysis involves studying a function's properties and behaviors. This helps to understand its characteristics, such as whether it's increasing, decreasing, or neither. In the context of determining if a function is one-to-one, we focus on its injectivity. To analyze functions, we often:

• Identify its domain and range
• Check for continuity
• Analyze its graph
• Investigate its algebraic properties

For a function to be one-to-one, it must map every input to a unique output. This means that it must pass the "horizontal line test" — a horizontal line should intersect the graph of the function at most once. If it does, the function is injective. This is a key step in function analysis when dealing with the concept of one-to-one functions.

Cubic Functions

Cubic functions are polynomial functions of degree three. They have the standard form of: \[ f(x) = ax^3 + bx^2 + cx + d \]Here, 'a', 'b', 'c', and 'd' are constants, with 'a' not equal to zero. These functions can have:

• One or three real roots
• A turning point(s)
• A point of inflection

The behavior of a cubic function as it approaches infinity or negative infinity is dictated by the leading term, x^3. Understanding this helps in analyzing its graph and behavior.A specific example, the function \( f(x) = x^3 \), doesn't have turning points. Its graph is symmetric about the origin and passes through the origin. It increases across its entire domain of all real numbers. It is also worth noting that because it is a strictly increasing function, \( f(x) = x^3 \) is automatically one-to-one.

Injective Functions

Injective functions are those where each element of the function's domain maps to a unique element in its range. This is also known as a "one-to-one" function. Mathematically, a function \( f(x) \) is injective if, for every pair of distinct elements \( a e b \) in the domain, we have \( f(a) e f(b) \) in the range.Testing for injectivity often involves solving the equation:\[ f(a) = f(b) \Rightarrow a = b \]For the example \( f(x) = x^3 \), we can see that:

• If \( a^3 = b^3 \), then \( a = b \)

This verification ensures us that the function is injective, as it meets the requirement where no two distinct inputs map to the same output. Such properties make injective functions an important class in mathematics, especially for establishing bijections and understanding the invertibility of functions.