Write A Function Formula For G Using The Function F

Deriving Function Formulas: Exploring the Relationship Between f(x) and g(x)

Understanding the relationship between two functions, specifically how one function, g(x), can be derived from another function, f(x), is a crucial concept in mathematics. This exploration delves into various methods and techniques for expressing g(x) as a transformation or combination of f(x). We'll cover a broad spectrum of transformations, including translations, reflections, stretches, compressions, and combinations thereof. Mastering these techniques is vital for success in calculus, algebra, and numerous applications in other scientific fields.

Understanding Function Transformations

Before we delve into specific examples, let's solidify our understanding of fundamental function transformations. These transformations alter the graph of a function without changing its inherent properties. The key transformations are:

1. Vertical Translations

A vertical translation shifts the graph of f(x) upwards or downwards. The formula for a vertical translation is:

g(x) = f(x) + k

where:

• k is a constant. If k is positive, the graph shifts upwards; if k is negative, it shifts downwards.

2. Horizontal Translations

A horizontal translation shifts the graph of f(x) to the left or right. The formula for a horizontal translation is:

g(x) = f(x - h)

where:

• h is a constant. If h is positive, the graph shifts to the right; if h is negative, it shifts to the left. Note the minus sign – this is counter-intuitive to many students initially.

3. Vertical Stretches and Compressions

A vertical stretch or compression scales the graph of f(x) vertically. The formula is:

g(x) = a * f(x)

where:

• a is a constant. If |a| > 1, the graph is stretched vertically; if 0 < |a| < 1, the graph is compressed vertically. If a is negative, there's also a reflection about the x-axis.

4. Horizontal Stretches and Compressions

A horizontal stretch or compression scales the graph of f(x) horizontally. The formula is:

g(x) = f(bx)

where:

• b is a constant. If |b| > 1, the graph is compressed horizontally; if 0 < |b| < 1, the graph is stretched horizontally. If b is negative, there's also a reflection about the y-axis.

5. Reflections

Reflections flip the graph of f(x) across an axis.

• Reflection across the x-axis: g(x) = -f(x)
• Reflection across the y-axis: g(x) = f(-x)

Combining Transformations

Often, g(x) is a result of multiple transformations applied to f(x). The order of operations is crucial. Generally, transformations are applied from the inside outwards: horizontal transformations (stretches, compressions, and translations) are performed before vertical transformations.

Example: Let's say f(x) = x², and g(x) = 2f(x - 3) + 1. This means:

1. Horizontal translation: The graph shifts 3 units to the right (x - 3).
2. Vertical stretch: The graph is stretched vertically by a factor of 2 (2f(...)).
3. Vertical translation: The graph shifts 1 unit upwards (... + 1).

Determining the Formula for g(x) from f(x) – A Step-by-Step Approach

Let's break down how to derive the formula for g(x) given f(x) and a description of the transformation.

Step 1: Identify the Transformations

Carefully analyze the transformation(s) applied to f(x) to obtain g(x). Note the direction and magnitude of each transformation (e.g., "shifted 2 units to the left," "stretched vertically by a factor of 3," "reflected across the x-axis").

Step 2: Apply the Transformation Formulas

Use the transformation formulas outlined above to express g(x) in terms of f(x). Remember the order of operations – horizontal transformations first, then vertical transformations.

Step 3: Simplify the Expression

Simplify the resulting expression to obtain the final formula for g(x).

Illustrative Examples

Let's work through a few detailed examples to solidify our understanding:

Example 1:

Given f(x) = x³, find the formula for g(x) if g(x) is obtained by shifting f(x) 2 units to the right and 1 unit down.

Solution:

1. Identify Transformations: Horizontal shift right by 2 units, vertical shift down by 1 unit.

2. Apply Transformation Formulas: The horizontal shift is represented by f(x - 2), and the vertical shift is represented by f(x) - 1. Combining these, we get:

   g(x) = f(x - 2) - 1

3. Substitute f(x): Substituting f(x) = x³, we get:

   g(x) = (x - 2)³ - 1

Example 2:

Given f(x) = √x, find the formula for g(x) if g(x) is obtained by stretching f(x) vertically by a factor of 3, reflecting it across the x-axis, and then shifting it 4 units to the left.

Solution:

1. Identify Transformations: Vertical stretch by a factor of 3, reflection across the x-axis, horizontal shift left by 4 units.

2. Apply Transformation Formulas: The vertical stretch is 3f(x), the reflection is -3f(x), and the horizontal shift is -3f(x + 4). Combining these, we have:

   g(x) = -3f(x + 4)

3. Substitute f(x): Substituting f(x) = √x, we get:

   g(x) = -3√(x + 4)

Example 3: A More Complex Scenario

Given f(x) = |x|, find the formula for g(x) if g(x) is obtained by compressing f(x) horizontally by a factor of 2, stretching it vertically by a factor of 4, reflecting it across the y-axis, and then shifting it 1 unit up.

Solution:

1. Identify Transformations: Horizontal compression by a factor of 2, vertical stretch by a factor of 4, reflection across the y-axis, vertical shift up by 1 unit.

2. Apply Transformation Formulas: The horizontal compression is f(2x), the vertical stretch is 4f(2x), the reflection across the y-axis is 4f(-2x), and the vertical shift is 4f(-2x) + 1. Combining these, we have:

   g(x) = 4f(-2x) + 1

3. Substitute f(x): Substituting f(x) = |x|, we get:

   g(x) = 4|-2x| + 1 = 8|x| + 1

Conclusion

Understanding function transformations is a fundamental skill in mathematics. By mastering the techniques outlined in this article, you can confidently derive the formula for g(x) from f(x) given any combination of translations, reflections, stretches, and compressions. Remember to carefully identify the transformations, apply the correct formulas in the correct order, and simplify the resulting expression. Practice is key to mastering this skill, so work through various examples and challenge yourself with increasingly complex scenarios. This understanding provides a solid foundation for more advanced concepts in mathematics and related fields.