Function Transformations Reflection Over Y-axis And Vertical Shifts

by ADMIN 68 views

When delving into the fascinating world of functions, understanding how transformations affect their properties is crucial. One such transformation is reflection across the y-axis. A critical question arises: Does reflecting a function over the y-axis preserve its domain and range? The answer, surprisingly, is not always straightforward and depends on the specific function in question.

Domains and Ranges: A Quick Review

Before we tackle the reflection, let's quickly recap what domains and ranges are. The domain of a function is the set of all possible input values (x-values) for which the function is defined. Think of it as the set of x-values you are allowed to plug into the function. The range, on the other hand, is the set of all possible output values (y-values) that the function can produce. It represents the y-values that result from plugging in the x-values from the domain.

Reflection Over the y-axis: The Transformation

Reflecting a function over the y-axis essentially creates a mirror image of the graph across the vertical y-axis. Mathematically, this transformation is achieved by replacing x with -x in the function's equation. So, if we have a function f(x), its reflection over the y-axis is given by f(-x). The key here is that every point (x, y) on the original graph is transformed to (-x, y) on the reflected graph. This change in the sign of the x-coordinate is what produces the mirror image.

When Domain and Range Remain the Same

The statement "When the function is reflected in the y-axis, the domain and range of the new function and the original are the same" is sometimes true, but not always. It holds true for functions that possess a certain type of symmetry: even functions.

An even function is defined as a function where f(x) = f(-x) for all x in its domain. This means that the function's output is the same whether you input x or -x. Graphically, even functions exhibit symmetry about the y-axis. If you were to fold the graph along the y-axis, the two halves would perfectly overlap. Classic examples of even functions include:

  • f(x) = x²: The parabola is symmetrical about the y-axis.
  • f(x) = cos(x): The cosine function is also symmetrical about the y-axis.
  • f(x) = |x|: The absolute value function is symmetrical about the y-axis.

For even functions, reflecting them across the y-axis results in the exact same graph. Since the graph doesn't change, neither does the domain nor the range. The original and reflected functions are indistinguishable.

When Domain and Range Change

However, not all functions are even. Odd functions, defined by the property f(-x) = -f(x), exhibit symmetry about the origin. This means that reflecting an odd function across the y-axis and then across the x-axis results in the original graph. Examples of odd functions include:

  • f(x) = x³: A cubic function.
  • f(x) = sin(x): The sine function.
  • f(x) = x: A linear function passing through the origin.

For odd functions, the reflection across the y-axis does change the function, but in a predictable way. The domain remains the same, but the range is reflected across the x-axis.

Even more importantly, many functions are neither even nor odd. For these functions, reflecting across the y-axis will generally change both the graph, the domain, and the range. Consider the simple linear function f(x) = x + 1. This function is neither even nor odd. Reflecting it across the y-axis gives us g(x) = f(-x) = -x + 1. The graphs of f(x) and g(x) are distinct lines, and while their ranges are both all real numbers, they are clearly different functions. If we consider a function such as f(x) = √x, the original domain is x ≥ 0, but the reflected function f(-x) = √(-x) has a domain of x ≤ 0. The range, however, remains the same.

Conclusion on y-axis reflection

In summary, whether the domain and range remain the same after reflection over the y-axis depends entirely on the function's symmetry. Even functions maintain their domain and range, while other functions may see changes in their domain, range, or both. Therefore, the statement is not universally true.

Transformations are the key to understanding the relationships between different functions. One of the most fundamental transformations is a vertical shift. Let's explore the statement: "The graph of the function g(x) = f(x) + k is a shift of k units down of the graph of f(x)." This statement requires careful consideration, as the direction of the shift depends critically on the sign of k.

Vertical Shifts Explained

A vertical shift involves moving the graph of a function up or down along the y-axis. The transformation g(x) = f(x) + k represents a vertical shift of the graph of f(x). The value of k dictates the magnitude and direction of the shift.

  • Positive k: If k is positive, the graph of f(x) is shifted upward by k units. This is because adding a positive constant to the output of the function increases each y-value by that amount, effectively moving the entire graph higher on the coordinate plane.
  • Negative k: If k is negative, the graph of f(x) is shifted downward by |k| units. Adding a negative constant to the output of the function decreases each y-value, resulting in a downward shift of the graph.

Analyzing the Statement

The original statement, "The graph of the function g(x) = f(x) + k is a shift of k units down of the graph of f(x)," is false because it only accounts for the case when k is negative. It omits the crucial detail that a positive k results in an upward shift. To make the statement accurate, it should be phrased more carefully.

Correcting the Statement

A more accurate way to describe the transformation is: "The graph of the function g(x) = f(x) + k is a vertical shift of the graph of f(x) by k units. If k is positive, the shift is upward; if k is negative, the shift is downward." Or, we could say, "The graph of the function g(x) = f(x) + k is a shift of |k| units up if k is positive and |k| units down if k is negative."

Examples of Vertical Shifts

Let's illustrate vertical shifts with a few examples:

  1. Consider the function f(x) = x².
    • If g(x) = f(x) + 2 = x² + 2, the graph of f(x) is shifted upward by 2 units.
    • If h(x) = f(x) - 3 = x² - 3, the graph of f(x) is shifted downward by 3 units.
  2. Consider the function f(x) = sin(x).
    • If g(x) = f(x) + 1 = sin(x) + 1, the sine wave is shifted upward by 1 unit.
    • If h(x) = f(x) - 0.5 = sin(x) - 0.5, the sine wave is shifted downward by 0.5 units.

Impact on Domain and Range

Vertical shifts have a direct impact on the range of a function but leave the domain unchanged. Since the transformation only moves the graph up or down, the set of possible x-values (the domain) remains the same. However, the set of possible y-values (the range) is shifted along with the graph.

  • If the original range of f(x) is [a, b], then the range of g(x) = f(x) + k will be [a + k, b + k]. This means that every value in the original range is increased by k.

Conclusion on Vertical Shifts

In conclusion, the statement that g(x) = f(x) + k is a shift of k units down of f(x) is inaccurate. A more precise description is that it represents a vertical shift of k units, with the direction (up or down) determined by the sign of k. Understanding vertical shifts is a fundamental building block for comprehending more complex function transformations. Vertical shifts affect the range of a function, and the domain remains unchanged.