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Magazine Chapter 3: Problem 16
Determine whether the function is one-to-one. $$ h(x)=x^{3}+8 $$
Short Answer
Expert verified
The function \( h(x) = x^3 + 8 \) is one-to-one.
Step by step solution
01
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.
02
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.
03
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.
04
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).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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:
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.
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 \( x \).
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.
