Problem 71 - 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. $$3 f\left(\frac{1}{4} x\right)$$

Step-by-Step Solution

Answer

Domain: [-4, 8], Range: [0, 9]

Step 1: Identify the Transformation Operations

The expression given is \( 3f\left(\frac{1}{4}x\right) \). There are two transformations applied: \( f\left(\frac{1}{4}x\right) \) is a horizontal stretch by a factor of 4, and \( 3f(x) \) is a vertical stretch by a factor of 3.

Step 2: Adjust the Domain for Horizontal Stretch

The original domain is \([-1, 2]\). The function \( f\left(\frac{1}{4}x\right) \) stretches the domain by a factor of 4. To find the new domain, multiply each endpoint of the original domain by 4. Thus, \(-1 \times 4 = -4\) and \(2 \times 4 = 8\), giving us the new domain \([-4, 8]\).

Step 3: Adjust the Range for Vertical Stretch

The original range is \([0, 3]\). The function \( 3f(x) \) stretches the range by a factor of 3. To find the new range, multiply each endpoint of the original range by 3. Thus, \(0 \times 3 = 0\) and \(3 \times 3 = 9\), giving us the new range \([0, 9]\).

Step 4: Combine Transformed Domain and Range

After applying the transformations, the new domain becomes \([-4, 8]\) and the new range becomes \([0, 9]\). This reflects the composition of both transformations on the function.

Key Concepts

Understanding Horizontal Stretch

A horizontal stretch affects how a function behaves along the x-axis, widening the graph out horizontally. It alters the way points are distributed over the x-axis by stretching or compressing them. For instance, when you have a function such as \( f\left(\frac{1}{4}x\right) \), it means you will stretch the x-values four times further apart.

Here’s what you need to remember about horizontal stretches:

• They change the domain of the function.
• The factor by which you multiply the x-coordinates is the reciprocal of what appears inside the function with x. So, \(\frac{1}{4}\) becomes 4 as the stretching factor.
• The original domain end-points of \([-1, 2]\) will be transformed by multiplying them by 4, widening them to \([-4, 8]\).

This means if you imagine the function’s original graph, it’s now broadened to cover a larger span on the horizontal axis.

Exploring Vertical Stretch

Vertical stretch involves the y-axis, influencing the output values of a function. It effectively pulls the graph upwards or downwards, amplifying or diminishing the effect. In the vertical stretch case, like \( 3f(x) \), each y-coordinate is three times as far from the x-axis than it was originally.

Key points to note about vertical stretches:

• They modify the range of the function.
• Every y-value is multiplied by the stretch factor. In our example, this factor is 3.
• The original range \([0, 3]\) scales up to \([0, 9]\) when multiplied by 3.

This ensures the function's graph reaches higher peaks and deeper valleys compared to the original.

Domain and Range Transformations

When functions undergo transformations, understanding how domain and range change is crucial. During a horizontal or vertical stretch, these transformations modify where and how the function plots on a graph.

Let’s break it down:

• **Domain Transformations**: These are related to horizontal changes. With a horizontal stretch by factor 4, each element of the domain \([-1, 2]\) is magnified, resulting in the new domain \([-4, 8]\).
• **Range Transformations**: Pertaining to vertical changes, such as multiplying a function by 3, which swells the range from \([0, 3]\) to \([0, 9]\).

By recognizing changes to domain and range, one can accurately graph transformations. Knowing each stretch factor helps identify the adjusted limits, giving a clearer picture of how the function behaves after transformation. It's like expanding a zoomed-in photo; you get more of the surrounding context in view.