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Chapter 4: Problem 15

Determine whether the function \(f\) is one-to-one. $$f(x)=\frac{1}{x}$$

Short Answer

Expert verified
The function \(f(x) = \frac{1}{x}\) is one-to-one.

Step by step solution

01

Understand the Definition

A function is one-to-one (injective) if every element of its range is mapped by at most one element of its domain. In other words, if \(f(a) = f(b)\) implies \(a = b\) for any elements \(a\) and \(b\) in the domain of \(f\).
02

Assume Equality for Outputs

Assume that \(f(a) = f(b)\) for some \(a\) and \(b\) in the domain of \(f(x) = \frac{1}{x}\). This means: \[ \frac{1}{a} = \frac{1}{b} \]
03

Simplify the Equation

To solve the equation \(\frac{1}{a} = \frac{1}{b}\), cross-multiply to eliminate the fractions: \[ b = a \] This shows that when the outputs are equal, the inputs must also be equal.
04

Conclusion Based on the Simplification

Since \(f(a) = f(b)\) implies \(a = b\), the function \(f(x) = \frac{1}{x}\) is one-to-one. Therefore, it satisfies the condition for being an injective function.

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Key Concepts

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

Injective Function
Functions are a fundamental part of mathematics, and an injective function (also known as a one-to-one function) possesses a special property. An injective function ensures that no two different inputs produce the same output. Therefore, each element in the function's range is mapped by a unique element from the domain. This can be confirmed with the simple rule: if \( f(a) = f(b) \), then \( a = b \).

In the provided solution, the function \( f(x) = \frac{1}{x} \) was analyzed to confirm this property. We assumed \( f(a) = f(b) \), leading to the equality \( \frac{1}{a} = \frac{1}{b} \). Solving this by cross-multiplying confirmed that \( a = b \), thus proving that the function is one-to-one.
  • An injective function never maps distinct elements in its domain to the same element in its range.
  • Injectivity can be visually verified if a horizontal line intersects the graph of the function at most once.
Function Range
The range of a function is the set of all possible outputs that the function can produce. When dealing with a function like \( f(x) = \frac{1}{x} \), determining the range helps us understand the extent of values the function can achieve.

For \( f(x) = \frac{1}{x} \), the range consists of all real numbers except zero. This is because the output can be any real number depending on the input, but it can never be zero, as no value of \( x \) satisfies \( \frac{1}{x} = 0 \).
  • The range provides a boundary of outputs, indicating all potential results based on different inputs within the domain.
  • Identifying the range can also help understand the behavior of the function across its domain.
Domain of a Function
Understanding the domain of a function is crucial for grasping where the function is defined and can be applied. The domain encompasses all input values for which the function is defined. For \( f(x) = \frac{1}{x} \), the function is undefined at \( x = 0 \) because division by zero is not possible.

Thus, the domain of \( f(x) = \frac{1}{x} \) is all real numbers except zero. This can be represented as \( (-\infty, 0) \cup (0, \infty) \).
  • The domain helps identify restrictions on input values that could lead to undefined behavior in functions.
  • Understanding the domain is key to correctly applying functions across different mathematical contexts.

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Most popular questions from this chapter

Studies relating serum cholesterol level to coronary heart disease suggest that a risk factor is the ratio \(x\) of the total amount \(C\) of cholesterol in the blood to the amount \(H\) of high-density lipoprotein cholesterol in the blood. For a female, the lifetime risk \(R\) of having a heart attack can be approximated by the formula $$R=2.07 \ln x-2.04 \quad \text { provided } \quad 0 \leq R \leq 1$$ For example, if \(R=0.65,\) then there is a \(65 \%\) chance that a woman will have a heart attack over an average lifetime. (a) Calculate \(R\) for a female with \(C=242\) and \(H=78\) (b) Graphically estimate \(x\) when the risk is \(75 \%\)

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