F(-x)

Definition

The function f(-x) is a transformation of the original function f(x) where the input variable x is replaced with its additive inverse, -x. This transformation reflects the function across the y-axis, resulting in a mirrored or flipped version of the original function.

5 Must Know Facts For Your Next Test

1. The transformation f(-x) reflects the original function f(x) across the y-axis, resulting in a mirrored or flipped version of the original function.
2. The graph of f(-x) is obtained by reflecting the graph of f(x) across the y-axis, keeping the same y-values but flipping the x-values.
3. The domain of f(-x) is the same as the domain of f(x), but the range may be different depending on the original function.
4. The transformation f(-x) is often used in conjunction with other transformations, such as translations and dilations, to create more complex transformations of functions.
5. The transformation f(-x) is particularly useful for modeling situations where the input variable represents a direction or orientation, such as in physics or engineering problems.

Review Questions

• Explain how the transformation f(-x) affects the graph of the original function f(x).
  • The transformation f(-x) reflects the graph of the original function f(x) across the y-axis. This means that the x-values of the graph are flipped or mirrored, while the y-values remain the same. The result is a graph that is a reflection or mirror image of the original function, with the same range but a potentially different domain depending on the properties of the original function.
• Describe the relationship between the original function f(x) and the transformed function f(-x).
  • The relationship between f(x) and f(-x) is that of an inverse function. Specifically, f(-x) is the reflection of f(x) across the y-axis, which means that the input variable x is replaced with its additive inverse, -x. This transformation effectively undoes the original function, resulting in a mirrored or flipped version of the graph. The domain and range of f(-x) may be different from the original function, depending on the properties of f(x).
• Analyze how the transformation f(-x) can be combined with other transformations to create more complex functions.
  • The transformation f(-x) can be combined with other transformations, such as translations, dilations, and reflections, to create more complex functions. For example, the transformation $f(-x + h)$ would first reflect the function across the y-axis and then translate it horizontally by $h$ units. Similarly, the transformation $a \cdot f(-x)$ would first reflect the function across the y-axis and then dilate it vertically by a factor of $a$. By combining these transformations, you can create a wide variety of modified functions that can be used to model more complex real-world scenarios.

Related terms

Reflection: A transformation that flips or mirrors a function across an axis, such as the x-axis or y-axis.

Inverse Function: A function that undoes the operation of another function, so that f(f^-1(x)) = x and f^-1(f(x)) = x.

Transformation of Functions: The process of modifying the graph of a function by applying various transformations, such as translations, reflections, stretches, and compressions.