Problem 57 - College Algebra with Corequisite Support

Question

Suppose the range of a function \(f\) is \([-5, \quad 8]\). What is the range of \(|f(x)| ?\)

Step-by-Step Solution

Answer

The range of \(|f(x)|\) is \([5, 8]\).

Step 1: Understand the Problem

We are given that the range of the function \( f \) is \([-5, 8]\). We need to find the range of \(|f(x)|\), the absolute value of \(f(x)\).

Step 2: Determine Effect of Absolute Value

The function \(|f(x)|\) will map any negative value of \(f(x)\) to its positive equivalent, while positive values remain unchanged. This means that the minimum value in \(|f(x)|\) will become positive if it was negative in \(f(x)\).

Step 3: Find Minimum of \(|f(x)|\)

The minimum value of \(f(x)\) is \(-5\) in the range \([-5, 8]\). Applying the absolute value, \(|-5| = 5\). This is the smallest positive value \(|f(x)|\) could take.

Step 4: Find Maximum of \(|f(x)|\)

The maximum value of \(f(x)\) is \(8\). Since it is already a positive number, the maximum value of \(|f(x)|\) is also \(8\).

Step 5: Construct the Range of \(|f(x)|\)

Combining the results from previous steps, the range of \(|f(x)|\) will be from the smallest to the largest value determined. Hence, the range of \(|f(x)|\) is \([5, 8]\).

Key Concepts

Absolute Value

Every function has values and some of them might be negative. The absolute value function transforms any number into its non-negative equivalent. For instance, if you have a number like \(-3\), its absolute value is \(|-3| = 3\). It essentially removes the negative sign. The absolute value follows a few important rules:

• For any positive number \(x\), the absolute value is \(x\) itself: \(|x| = x\).
• For any negative number \(-x\), the absolute value flips it to be positive: \(|-x| = x\).
• Zero is its own absolute value: \(|0| = 0\).

In our context, since the original function \(f(x)\) has a range of \([-5, 8]\), applying the absolute value function means flipping the \(-5\) to \(5\), while leaving all positive values unchanged.

Function Transformation

The concept of function transformation involves making changes to the basic form of a function, such as shifting, stretching, or reflecting it. One common transformation is the absolute value transformation, which affects the appearance and behavior of a function's graph. When you apply the absolute value to a function \(f(x)\), every negative portion of the graph gets flipped up to the positive side. This creates a "V" shape if the function is linear.

• Horizontal shifts move the graph left or right.
• Vertical shifts move it up or down.
• Reflections across axes are also part of transformations.

For the function \(|f(x)|\), this transformation changes how we interpret the range, impacting any negative values to ensure they are presented positively.

Range Determination

Determining the range of a function is about identifying all the possible values that the function's output can take. In simpler terms, the range tells us the span of the y-values over which the function is defined. For the function \(|f(x)|\), knowing the changes introduced by the absolute value is key. For example:

• If the original range of \(f(x)\) was \([-5, 8]\), applying the absolute value modifies the lower bound.
• The new range reflects this transformation and becomes \([5, 8]\).
• This adjustment ensures that all potential results of \(|f(x)|\) are non-negative, following the absolute value's effect.

Recognizing how transformations influence the range is crucial to solving problems involving absolute value functions.