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Magazine Chapter 6: Problem 24
Decide whether each function is one-to-one. $$f(x)=-7$$
Short Answer
Expert verified
The function \( f(x) = -7 \) is not one-to-one.
Step by step solution
01
Understanding the Problem
We need to determine if the function \( f(x) = -7 \) is a one-to-one function. A one-to-one function is where each input maps to a unique output, and each output is mapped by exactly one input.
02
Analyzing the Function
The function \( f(x) = -7 \) is a constant function, meaning that for any value of \( x \), the output is always \(-7\).
03
Examining One-to-One Criteria
For a function to be one-to-one, any given output value must correspond to exactly one input. In the function \( f(x) = -7 \), every possible input \( x \) results in the same output \(-7\), violating this one-to-one correspondence.
04
Conclusion
Since the same output is produced for every input, the function \( f(x) = -7 \) is not one-to-one.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Constant Functions
Constant functions are a unique type of mathematical function where the output is the same no matter what the input is. In other words, no matter what value of \( x \) you choose, the function will always return the same result. For example, in the function \( f(x) = -7 \), the output will always be \(-7\) regardless of the input value.
- These types of functions are represented graphically as horizontal lines on the Cartesian plane.
- They are simple yet significant in helping to understand more complex functions.
Function Analysis
When analyzing functions, we examine their properties and behavior to gain a better understanding. Analyzing a constant function like \( f(x) = -7 \) is straightforward since it outputs the same value irrespective of input. Here's what to look at:
- **Purpose:** Understand why this function is constant and what it represents.
- **Output Consistency:** All input values yield the same output (e.g., always \(-7\)).
- **Graphical Representation:** As a straight horizontal line, signifying no change in \( y \) when \( x \) changes.
Function Mapping
Function mapping is critical when determining the nature of a function. It describes how each element of one set (the domain) is paired with an element in another set (the range).With constant functions like \( f(x) = -7 \), all elements in the domain map to a single point in the range (\(-7\)). This is seen as:
- Every possible input \( x \) in the domain results in the same output \(-7\).
- Creates a many-to-one relationship, illustrating it's not one-to-one.
Math Problem-Solving
In math problem-solving, breaking down exercises into manageable steps is essential. Solving whether a function is one-to-one involves understanding its definition and mechanics.Here's a structured approach to tackle problems like this:
- **Identify the Function Type:** Detect if it's constant, linear, or another type. Constant functions like \( f(x) = -7 \) are easy to identify.
- **Assess Output Versatility:** Determine if each input results in a unique output. For constant functions, the outputs are always the same regardless of input.
- **Apply Criteria:** Use insights from function mapping to check if the one-to-one condition is met. Constant functions typically don't satisfy this.
