Problem 55
Question
If \(f(x)\) is one-to-one, can anything be said about \(g(x)=-f(x) ?\) Is it also one-to-one? Give reasons for your answer.
Step-by-Step Solution
Verified Answer
Yes, \( g(x) = -f(x) \) is one-to-one because it retains the distinct outputs for distinct inputs like \( f(x) \).
1Step 1: Understanding One-to-One Functions
A function \( f(x) \) is considered one-to-one if for every pair of distinct inputs \( x_1 \) and \( x_2 \), the outputs \( f(x_1) \) and \( f(x_2) \) are also distinct. This means \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
2Step 2: Analyzing Function Modification
The function \( g(x) = -f(x) \) represents a transformation of \( f(x) \) by reflecting its outputs over the x-axis. We need to verify if this transformation preserves the one-to-one nature of \( f(x) \).
3Step 3: Proving One-to-One Property for g(x)
To determine if \( g(x) \) is one-to-one, assume \( g(x_1) = g(x_2) \). Plug in \( g(x) = -f(x) \) to get \( -f(x_1) = -f(x_2) \). Simplify this to find \( f(x_1) = f(x_2) \). Since \( f(x) \) is one-to-one, \( x_1 = x_2 \) must hold, confirming \( g(x) \) is also one-to-one.
4Step 4: Conclusion
Since the transformation \( g(x) = -f(x) \) retains the distinctness of outputs for distinct inputs due to \( f(x) \) being one-to-one, \( g(x) \) is confirmed to be a one-to-one function.
Key Concepts
Function TransformationReflection Over X-axisDistinct Inputs and OutputsPreserving Bijectivity
Function Transformation
In mathematics, a function transformation refers to the process of changing a function's appearance or position on a coordinate plane. This can include shifts, stretches, compressions, and reflections. When performing a transformation on a function, it's crucial to understand how these changes affect the original function's properties.
For instance, if we have a function \( f(x) \) and we apply a transformation to create a new function \( g(x) = -f(x) \), we're reflecting every point of \( f(x) \) over the x-axis. This specific transformation is a type of reflection. While the shape of the graph changes, it's key to analyze whether essential characteristics, like being one-to-one, are maintained. This is important because it helps determine if the new function still behaves similarly to the original in terms of unique outputs for given inputs.
For instance, if we have a function \( f(x) \) and we apply a transformation to create a new function \( g(x) = -f(x) \), we're reflecting every point of \( f(x) \) over the x-axis. This specific transformation is a type of reflection. While the shape of the graph changes, it's key to analyze whether essential characteristics, like being one-to-one, are maintained. This is important because it helps determine if the new function still behaves similarly to the original in terms of unique outputs for given inputs.
Reflection Over X-axis
A reflection over the x-axis is a specific type of function transformation. It involves flipping the graph of a function such that all points are mirrored with respect to the x-axis. This means if a point on the original function is at \((x, y)\), then on the reflected function it will be at \((x, -y)\).
In the context of our given function \( g(x) = -f(x) \), the negative sign indicates this reflection. It's important to note that reflection, in general, does not alter the relationship between distinct inputs and outputs, if the original function was one-to-one. The transformation only affects the graph's orientation. Thus, the one-to-one nature of a function, meaning each input corresponds to a unique output, remains preserved through this reflection.
In the context of our given function \( g(x) = -f(x) \), the negative sign indicates this reflection. It's important to note that reflection, in general, does not alter the relationship between distinct inputs and outputs, if the original function was one-to-one. The transformation only affects the graph's orientation. Thus, the one-to-one nature of a function, meaning each input corresponds to a unique output, remains preserved through this reflection.
Distinct Inputs and Outputs
Distinct inputs and outputs are fundamental to the concept of a one-to-one function. A function \( f(x) \) is one-to-one if it never assigns the same output to two different inputs. This condition is mathematically represented as: if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).
For our transformed function \( g(x) = -f(x) \), preserving this distinctness is crucial. When checked, even after reflecting \( f(x) \) over the x-axis, the distinct nature of outputs given distinct inputs holds steady. This is because reflection over the x-axis does not merge outputs; it merely inverts them vertically on a coordinate plane without loss of uniqueness.
For our transformed function \( g(x) = -f(x) \), preserving this distinctness is crucial. When checked, even after reflecting \( f(x) \) over the x-axis, the distinct nature of outputs given distinct inputs holds steady. This is because reflection over the x-axis does not merge outputs; it merely inverts them vertically on a coordinate plane without loss of uniqueness.
- Ensures that for any two differing inputs, their outputs will remain distinct even after transformation.
- The function's property of mapping each specific input to a unique output continues unabated.
Preserving Bijectivity
Bijectivity is a property describing a function that is both one-to-one (injective) and onto (surjective). However, when we focus on preserving bijectivity in the context of transformations like\( g(x) = -f(x) \), we primarily deal with maintaining the injective nature.
For a one-to-one function \( f(x) \), reflecting over the x-axis does not alter the relationship between inputs and outputs. The invertible nature of the function remains intact, confirming that the transformation preserves bijectivity concerning being one-to-one.
For a one-to-one function \( f(x) \), reflecting over the x-axis does not alter the relationship between inputs and outputs. The invertible nature of the function remains intact, confirming that the transformation preserves bijectivity concerning being one-to-one.
- Injective (one-to-one): No two distinct inputs map to the same output, before or after transformation.
- Surjective (onto): Not directly addressed by reflection alone but important to full bijectivity if overall mapping is sustained.
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