Problem 67 - 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: \([-1, 2]\); Range: \([-3, 0]\)

Step 1: Identify the Domain of \\(-f(x)\\)

The domain of a function is the set of all possible input values. Since we are given that the domain of \(f(x)\) is \([-1, 2]\), the domain of \(-f(x)\) remains the same, which is \([-1, 2]\).

Step 2: Determine the Effect on the Range

To determine how the range of \(-f(x)\) is affected, note that negating a function will invert its output around the horizontal axis. For \(f(x)\) with a range \([0, 3]\), its negation \(-f(x)\) will have its range as \([-3, 0]\).

Step 3: Write the Domain and Range of \\(-f(x)\\)

Now, summarize the findings. The domain of \(-f(x)\) is the same as \(f(x)\), which is \([-1, 2]\). The range is the result of mirroring the original range \([0, 3]\) across the x-axis, resulting in \([-3, 0]\).

Key Concepts

Negation of Functions

When we talk about the negation of functions, we mean flipping the output values of the function across the horizontal axis. Imagine a simple smiley face drawn on the y-axis. By negating the function, this smiley turns into a frown because every y-value turns into its opposite; positive values become negative, and vice versa.

In mathematical terms, if you have a function \(f(x)\), then the negated function is represented as \(-f(x)\). This operation impacts the range of the function but leaves the domain unaffected. The domain does not change because negation is about output, not input.

For example, if the original range of \(f(x)\) is \( [0, 3] \), then the range of the negated function \(-f(x)\) will be \( [-3, 0] \). This is because we simply take the negative of each value in the original range.

Function Transformation

Function transformation involves changing a function's position on the graph without altering its basic shape. Negation is one type of transformation, specifically reflecting a function around an axis.

Alongside negation, transformations can occur through:

• Vertical shifts (moving up or down)
• Horizontal shifts (moving left or right)
• Rescaling (stretching or compressing)

In the case of \(-f(x)\), reflecting across the x-axis illustrates a transformation, while keeping the domain the same. Transformations like these do not add or remove any input values, but rather adjust how those inputs relate to their corresponding outputs. This is critical in understanding how functions can be manipulated graphically without changing their fundamental behavior.

Input and Output Values

In functions, input values are essential as they determine what the function uses to produce an output. The set of all possible input values is known as the domain. For the function \(f(x)\), the domain is \([-1, 2]\), meaning the function can accept any value within this range.

Output values, on the other hand, are the results of these input values being processed through the function. They make up what is called the range of the function. For \(f(x)\), the outputs originally spread across the range \([0, 3]\).

When a function like \(-f(x)\) is negated, its output values are inverted around the x-axis. Consequently, while the input values (or domain) stay the same, the output values (or range) change to \([-3, 0]\), emphasizing the connection between an input and the transformed output.