Problem 16 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. $$ h(x)=x^{3}+8 $$

Step-by-Step Solution

Answer

The function \( h(x) = x^3 + 8 \) is one-to-one.

Step 1: Identify Basic Function Type

The function given is \( h(x) = x^3 + 8 \). The basic form is \( f(x) = x^3 \), which is a cubic function. Cubic functions have no sharp turns or breaks, indicating a smooth, continuous curve.

Step 2: Understand One-to-One Functions

A function is one-to-one if and only if every y-value has a unique x-value, or equivalently, if it passes the horizontal line test, meaning no horizontal line intersects the graph more than once.

Step 3: Apply Horizontal Line Test

For the function \( h(x) = x^3 + 8 \), consider horizontal lines \( y = c \). Since \( x^3 \) is a strictly increasing function, the graph of \( h(x) = x^3 + 8 \) will pass the horizontal line test due to its continuous, ever-increasing nature.

Step 4: Mathematical Justification

To confirm, assume \( h(x_1) = h(x_2) \). This implies \( x_1^3 + 8 = x_2^3 + 8 \). Simplifying gives \( x_1^3 = x_2^3 \) which implies \( x_1 = x_2 \) since the cube function is strictly monotonic (always increasing or decreasing).

Step 5: Conclusion

Since the function \( h(x) = x^3 + 8 \) passes the horizontal line test and no value repeats for different x-values, the function is indeed one-to-one.

Key Concepts

Cubic Function

A cubic function is a type of polynomial function where the highest degree of the variable is three. It can be expressed in the general form of \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and \( a eq 0 \). In our example, the cubic function is \( h(x) = x^3 + 8 \). Here, the term \( x^3 \) ensures that it is a cubic function, while the constant 8 simply shifts the entire graph upward.

Key characteristics of cubic functions include:

• Continuity: These functions are continuous for all real numbers, meaning there are no breaks or holes in the graph.
• Symmetry: They are often symmetric in shape. Specifically, when there are no quadratic or linear components, it has origin symmetry, as seen in \( f(x) = x^3 \).
• Inflection Point: Cubic functions typically have one point of inflection, where the curvature of the graph changes direction.

Understanding the behavior and shape of cubic functions aids in analyzing their properties, such as whether they are one-to-one.

Horizontal Line Test

The horizontal line test is a method used to determine if a function is one-to-one. Essentially, it checks if any horizontal line drawn across the graph of a function intersects the graph more than once. Passing this test means each y-value corresponds to only one x-value, confirming the function is one-to-one.

Here's how to apply the horizontal line test:

• Draw a horizontal line (any line where \( y = c \)).
• Look for intersections:
  • If the line intersects the graph of the function more than once, the function is not one-to-one.
  • If it only intersects once, the function is one-to-one.

For \( h(x) = x^3 + 8 \), because the core of the function, \( x^3 \), is a strictly increasing function, any horizontal line will intersect it only once. Thus, it passes the horizontal line test, proving the function is one-to-one.

Strictly Monotonic

A function is said to be strictly monotonic if it is either entirely non-increasing or non-decreasing. In simpler terms, it means that as one moves along the graph, the function consistently rises or falls but never both.

The function \( h(x) = x^3 + 8 \) demonstrates this property by being strictly increasing:

• Strictly Increasing: As the input \( x \) increases, the output \( h(x) \) also increases. This is true for the base function \( x^3 \) as its slope is always positive for all values of \( x \).

This characteristic is a key reason behind the function being one-to-one, because in strictly monotonic functions different input values always lead to different output values.

Understanding strictly monotonic behavior helps to ensure that no two different x-values will yield the same y-value, solidifying the function's one-to-one nature.