Problem 73 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Domain: \([-\frac{1}{2}, 1]\); Range: \([0, 3]\)

Step 1: Write the Function Transformation

The function given is transformed as \( f(2x) \). Here, the transformation involves a horizontal compression by a factor of 2.

Step 2: Determine the New Domain

The original domain is \([-1, 2]\). When compressing horizontally by a factor of 2, we solve \( 2x \) in the interval, resulting in the inequality \(-1 \, \leq \, 2x \, \leq \, 2 \). Dividing everything by 2 gives \(-\frac{1}{2} \, \leq \, x \, \leq \, 1 \). Thus, the new domain is \([-\frac{1}{2}, 1]\).

Step 3: Assess Vertical Range Impact

Horizontal transformations do not affect the range of the function. Therefore, the range remains the same as the original, which is \([0, 3]\).

Key Concepts

Horizontal Compression

When you encounter the transformation of a function such as \( f(2x) \), you're dealing with a horizontal compression. This concept means that the function's graph is compressed towards the y-axis. The horizontal compression factor is given by the number that multiplies \(x\), in this case, 2. Let's break it down:

• If the function is \(f(x)\), when it becomes \(f(2x)\), each x-value is halved because you're multiplying the variable by 2.
• Visualize it as "squashing" the graph horizontally, bringing points closer to the y-axis.

Horizontal compression affects only the x-values of the domain, not the y-values in the range. That's why it only transforms the domain while the range remains unchanged.

Domain and Range

Understanding the domain and range is crucial for any function transformation. The domain of a function refers to all the possible x-values that can be inputted into the function, while the range refers to all the possible y-values the function can output.When a function undergoes a horizontal compression, the domain changes according to the compression factor. In our example, the function \(f(x)\) originally had a domain of \([-1, 2]\).

• With a horizontal compression by a factor of 2 (as in \(f(2x)\)), we calculate the new domain by transforming each limit: solving the inequality gives \(-\frac{1}{2} \leq x \leq 1\).
• The range remains unaffected by horizontal transformations, so it stays \([0, 3]\).

It's essential to carefully calculate and verify these transformations to ensure the domain and range are correctly determined.

Inequalities

Inequalities play a key role in determining the new domain during transformations. When transforming \(f(x)\) to \(f(2x)\), we adjust the domain by applying inequalities.Here's how it's done:

• Start with the original domain given as an interval: \([-1, 2]\).
• Apply the transformation to the domain: evaluate \(2x\) within the original limits, \(-1 \leq 2x \leq 2\).
• Solve these inequalities by dividing each part by 2 to retrieve the expression for \(x\): \(-\frac{1}{2} \leq x \leq 1\).

Inequalities help us understand how transformations affect the specific limits of the domain. By dividing appropriately in the case of compression or stretching, we ensure the function's new domain is correctly described.