Problem 73 - A Graphical Approach to College Algebra

Question

Let the domain of \(f(x)\) be \([-1,2]\) and the range be \([0,3] .\) Find the domain and range of the following. $$f(-x)$$

Step-by-Step Solution

Answer

Domain: [-2, 1]; Range: [0, 3].

Step 1: Understanding the Function Transformation

When we have a function of the form \( f(x) \) and apply \( f(-x) \), it represents a reflection of the graph of \( f(x) \) across the y-axis. This transformation affects the domain of the function but does not affect the range.

Step 2: Finding the New Domain

Initially, the domain of \( f(x) \) is \([-1, 2]\). When the function becomes \( f(-x) \), all \( x \) values are negated. Reflect the domain across the y-axis: -1 becomes 1 and 2 becomes -2. Thus, the modified domain would be \([-2, 1]\).

Step 3: Finding the New Range

The range describes the set of possible output values. Since reflecting over the y-axis does not change these values, the range of \( f(-x) \) remains the same as \( f(x) \), which is \([0, 3]\).

Key Concepts

Domain and Range

The domain and range of a function are essential to understand how the function behaves in terms of inputs and outputs. Let’s break it down:

• Domain: The domain of a function is the complete set of possible input values (usually denoted as \(x\)), while the range is the complete set of possible output values (often denoted as \(y\)).
• Example: For a function \(f(x)\) with a domain of \([-1, 2]\), this means \(x\) can take any value between -1 and 2, inclusive.

When applying a transformation like \(f(-x)\), it impacts the domain because it reflects the graph across the y-axis. However, the range remains unchanged. This is because reflection across the y-axis only affects the input values, not the heights (or output values) of the graph.

• In our example, the original domain \([-1, 2]\) changes to \([-2, 1]\) after reflection across the y-axis, because each point \(x\) becomes \(-x\).
• The range remains \([0, 3]\), as these are the possible outputs the function can achieve regardless of the transformation.

Thus, understanding domains and ranges is crucial when graphing and transforming functions, as they show you the boundaries of what your function will do.

Reflection Across the Y-axis

Reflection across the y-axis is a specific type of function transformation that can seem tricky at first but is actually quite straightforward once you get the hang of it.When a function changes from \(f(x)\) to \(f(-x)\), every point on the graph is mirrored over the y-axis. This reflection means:

• Every positive \(x\) value swaps its sign to become negative.
• Every negative \(x\) value swaps its sign to become positive.

This change affects the domain of the function, but as mentioned before, it does not influence the range. So if initially, the graph of the function extended from \(-1\) to \(2\) on the \(x\)-axis, after reflection, it will extend from \(-2\) to \(1\). This results from flipping each \(x\) value across the y-axis, essentially reversing its sign.However, the reflection does not alter the height achieved by these points, thus the range remains constant. Recognizing this transformation helps significantly in visualizing and understanding how functions will appear when graphed after such a shift.

Function Graphing

Graphing a function is all about visually interpreting the behavior of the function based on its properties, like domain and range, and transformations such as reflections.When you graph \(f(x)\), you're plotting points based on input \(x\) and output \(f(x)\). To maintain accuracy, consider:

• Domain: Graph the points only within the domain of the function. For \(f(x)\) from \([-1, 2]\), this means plotting for \(x\) values between -1 and 2.
• Range: Make sure the y-values (outputs) remain within the range from \([0, 3]\).

When graphing \(f(-x)\), simply take the original graph of \(f(x)\) and reflect it across the y-axis.

• Every point will now be graphed with \(-x\) instead of its original \(x\) coordinate.
• This gives you a visual mirror image of the original function but stays within the set range \([0, 3]\).

Function graphing, especially with transformations like reflection, helps deepen your comprehension of how functions behave under various modifications. By routinely practicing these concepts, you enhance your ability to manipulate and understand complex functions.