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Chapter 2: Problem 20

Determine whether the function is one-to-one. $$f(x)=x^{4}+5, \quad 0 \leq x \leq 2$$

Short Answer

Expert verified
The function is one-to-one on the interval \(0 \leq x \leq 2\).

Step by step solution

01

Understanding the Function

The given function is \( f(x) = x^4 + 5 \) and is defined for the domain \( 0 \leq x \leq 2 \). Our task is to determine whether this function is one-to-one over this interval.
02

Definition of a One-to-One Function

A function is considered one-to-one (injective) if and only if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \) for every \( x_1, x_2 \) within the domain.
03

Consider the Nature of the Function

Look at the expression \( x^4 + 5 \). Since \( (x^4 + 5) \) is strictly increasing for \( x \geq 0 \), it is more likely to be one-to-one between \( x = 0 \) and \( x = 2 \).
04

Calculating the Derivative

Take the derivative \( f'(x) \) to find where the function is increasing. We have: \[ f'(x) = 4x^3 \]
05

Analyze the Derivative

For \( 0 \leq x \leq 2 \), the derivative \( f'(x) = 4x^3 \geq 0 \). Since \( f'(x) > 0 \) for \( x > 0 \), the function is strictly increasing in the domain.
06

Conclusion about One-to-One Nature

Since the function is strictly increasing throughout the entire interval from 0 to 2, it means that \( f(x_1) = f(x_2) \) only if \( x_1 = x_2 \). Therefore, the function is one-to-one over the given interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding One-to-One Functions
Determining if a function is one-to-one is crucial in various mathematical applications. A one-to-one function, also known as an injective function, has a distinct behavior. If you imagine graphically picking any two different points
  • Within the function's domain, these points must map to distinct outputs on the range.
  • This means if the outputs are the same, the inputs must also be the same.
The formal mathematical way to express this is: If \( f(x_1) = f(x_2) \), then it must be true that \( x_1 = x_2 \). This distinct feature helps avoid any repetition in outputs, making the function injective. Understanding one-to-one functions assists in verifying if inverses exist for those functions.
Exploring the Derivative
The derivative of a function gives us vital information about how the function behaves. It tells us the rate at which the function's output changes with respect to changes in input. For a function like \( f(x) = x^4 + 5 \), calculating the derivative, which is \( 4x^3 \),
  • Helps determine where the function is increasing or decreasing.
  • A positive derivative suggests that the function is increasing at that point.
  • On the contrary, a negative derivative would indicate a decreasing function.
By examining the derivative over a given interval, we can infer much about the nature of the function, like its one-to-one property or if it's monotonic (entirely increasing or decreasing).
Identifying Increasing Functions
An increasing function maintains a continuous upward trend as you move from left to right along the x-axis. For the function \( f(x) = x^4 + 5 \),
  • The calculated derivative \( f'(x) = 4x^3 \) is greater than zero for all \( x > 0 \).
  • This positivity signifies that the function continuously rises as \( x \) moves from 0 to 2.
In simple terms, this ensures that for any two numbers, \( x_1 \) and \( x_2 \) within this interval, if \( x_1 < x_2 \), then \( f(x_1) < f(x_2) \). Thus, the function is monotonically increasing, which implies that it is also a one-to-one function over this interval.
Understanding Injective Functions
Injective functions are a special class of functions where each output is uniquely matched to one input. That essentially differentiates them from other types, like non-injective or surjective functions. When a function is described as injective,
  • It's implying that no two distinct inputs will produce the same output.
  • This uniqueness is important when determining if the function can be reversed (as in finding an inverse function).
With the function \( f(x) = x^4 + 5 \), its derivative confirming it as increasing proves its injective nature over the interval [0, 2]. Ensuring this property guarantees that you can retrace from output back to the original input without ambiguity.

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Most popular questions from this chapter

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