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Magazine Chapter 2: Problem 19
Determine whether the function is one-to-one. $$f(x)=\frac{1}{x^{2}}$$
Short Answer
Expert verified
The function is not one-to-one.
Step by step solution
01
Understanding One-to-One Functions
A function is one-to-one if it assigns a unique output for every unique input and vice versa, meaning no two different inputs map to the same output.
02
Checking Outputs for Inputs
Consider two different inputs, say, \( x_1 \) and \( x_2 \), where \( x_1 eq x_2 \). We will evaluate if \( f(x_1) = f(x_2) \) could result in \( x_1 = x_2 \), ensuring the function's one-to-one property is met.
03
Function Evaluation
Evaluate the function for both inputs: \( f(x_1) = \frac{1}{x_1^2} \) and \( f(x_2) = \frac{1}{x_2^2} \). Suppose \( f(x_1) = f(x_2) \); then: \[ \frac{1}{x_1^2} = \frac{1}{x_2^2} \]
04
Solving for Equality
Solving \( \frac{1}{x_1^2} = \frac{1}{x_2^2} \), we get \( x_1^2 = x_2^2 \), which implies \( x_1 = x_2 \) or \( x_1 = -x_2 \). Since \( x_1 eq x_2 \), \( x_1 = -x_2 \) must also be considered.
05
Conclusion
The condition \( x_1 = -x_2 \) means different values of \( x_1 \) can lead to the same output \( f(x) \). Therefore, \( f(x) \) is not one-to-one because more than one input can lead to the same output.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Evaluation
Function evaluation involves taking a specific input value for a function and determining the corresponding output value. In simpler terms, to evaluate a function means to substitute the input variable with a specific number. For instance, in the function \( f(x) = \frac{1}{x^2} \), if you were to evaluate it at \( x = 3 \), you would substitute 3 into the function to get \( f(3) = \frac{1}{3^2} = \frac{1}{9} \). This process is essential because it shows us the behavior of the function for particular inputs.
Some key points to remember about function evaluation are:
Some key points to remember about function evaluation are:
- Always substitute the input directly into the function.
- Check your calculations to ensure accuracy.
- By evaluating at multiple points, you gain insights into the function's behavior.
Solving Equations
Solving equations allows us to determine values that satisfy a given mathematical expression. In the context of one-to-one functions, solving equations helps to verify whether different inputs can produce the same output, which would mean the function is not one-to-one.
Let鈥檚 look at the function \( f(x) = \frac{1}{x^2} \). To check its one-to-one nature, compare the outputs for two distinct inputs, \( x_1 \) and \( x_2 \), solving \( \frac{1}{x_1^2} = \frac{1}{x_2^2} \). This equation simplifies to \( x_1^2 = x_2^2 \), which provides solutions like \( x_1 = x_2 \) or \( x_1 = -x_2 \).
Some crucial points for solving such equations include:
Let鈥檚 look at the function \( f(x) = \frac{1}{x^2} \). To check its one-to-one nature, compare the outputs for two distinct inputs, \( x_1 \) and \( x_2 \), solving \( \frac{1}{x_1^2} = \frac{1}{x_2^2} \). This equation simplifies to \( x_1^2 = x_2^2 \), which provides solutions like \( x_1 = x_2 \) or \( x_1 = -x_2 \).
Some crucial points for solving such equations include:
- Look for all possible solutions, even if they seem contradictory at first.
- Ensure that we consider both positive and negative roots.
- Double-check solutions to verify they are logical within the context of the problem.
Uniqueness of Outputs
The uniqueness of outputs is a defining feature of one-to-one functions. A function is one-to-one if every input corresponds to a unique output. In other words, no two different inputs should map to the same output.
In the function \( f(x) = \frac{1}{x^2} \), we saw that \( x_1 \) and \( -x_1 \) produce the same output, thus, the function does not have a unique mapping from inputs to outputs. It illustrates that two distinct values of \( x \) (like 3 and -3) yield the same function value \( \frac{1}{9} \).
Here are some clarifications about ensuring uniqueness:
In the function \( f(x) = \frac{1}{x^2} \), we saw that \( x_1 \) and \( -x_1 \) produce the same output, thus, the function does not have a unique mapping from inputs to outputs. It illustrates that two distinct values of \( x \) (like 3 and -3) yield the same function value \( \frac{1}{9} \).
Here are some clarifications about ensuring uniqueness:
- Use the horizontal line test graphically: if a horizontal line intersects the graph more than once, the function is not one-to-one.
- Mathematically show that different inputs do not yield the same outputs.
- Understanding that a function mapping different inputs to the same output cannot be one-to-one.
Precalculus
Precalculus serves as a bridge between algebra and calculus, involving the study of functions, complex numbers, and vectors among other topics. Understanding functions, especially the concept of one-to-one functions, is crucial in precalculus as it lays the foundation for more advanced topics.
This exercise on determining one-to-one functions introduces crucial precalculus ideas:
This exercise on determining one-to-one functions introduces crucial precalculus ideas:
- Understanding the definition and properties of different types of functions.
- Developing the skills to solve and manipulate equations to draw conclusions about function properties.
- Applying logical reasoning to solve mathematical problems, preparing students for calculus.
