Problem 119 - A Graphical Approach to College Algebra

Question

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=|x-6|$$

Step-by-Step Solution

Answer

Restrict the domain to \( x \geq 6 \) to make \( f(x) = |x-6| \) one-to-one without changing the range.

Step 1: Understand the Function

The given function is \( f(x) = |x-6| \). This is an absolute value function, which creates a "V" shape when graphed. The vertex of this function is at \( x = 6 \).

Step 2: Determine the One-to-One Condition

A function is one-to-one if each input corresponds to exactly one output. Since the absolute value function is symmetric around its vertex, it is not one-to-one over its entire domain because, for instance, \( f(5) = 1 \) and \( f(7) = 1 \).

Step 3: Restrict the Domain

To make the function one-to-one, we need to consider only one side of the symmetry. We can restrict the domain to \( x \geq 6 \) or \( x \leq 6 \). Either option will help maintain a one-to-one condition, and the range, \( [0, \infty) \), will not change.

Step 4: Choose a Practical Domain Restriction

Choosing the restriction \( x \geq 6 \), the function becomes increasing, which is simple to work with and clearly establishes a one-to-one relationship.

Step 5: Verify the Range

With \( x \geq 6 \), every \( y \) value in the range \( [0, \infty) \) is still achievable. For example, \( f(6) = 0 \) and for any \( x > 6 \), \( f(x) > 0 \). Thus, the range remains \([0, \infty)\).

Key Concepts

Absolute Value Function

An absolute value function is defined by the expression \( f(x) = |x-a| \), where \( a \) is a constant. This function turns any negative outcomes of \( x-a \) into positive values, hence the "absolute" part. The graph of an absolute value function typically appears as a "V" shape.

The point at the bottom of the "V" is called the vertex, and it occurs at \( x = a \). For \( f(x) = |x-6| \), the vertex is at \( x = 6 \). This point represents the minimum value of the function, which is 0.

Absolute value functions are often used to describe situations where only the size of a number is important, not its direction.

One-to-One Function

A one-to-one function has a unique output for every unique input, meaning no two different inputs have the same output. For the absolute value function, this is not naturally the case.

Due to its symmetry about its vertex, each output is repeated once on either side of the vertex. For instance, with \( f(x) = |x-6| \), both \( x = 5 \) and \( x = 7 \) produce the same result, \( f(x) = 1 \).

To make an absolute value function one-to-one, we restrict the domain so that the function is either entirely increasing or decreasing.

Function Range

The range of a function refers to the set of possible outputs. For the absolute value function \( f(x) = |x-6| \), the range naturally is \([0, \infty)\).

This means that the outputs start from 0 (at \( x = 6 \)) and extend to positive infinity as \( x \) moves away from 6 in either direction.

Even when we restrict the domain to make the function one-to-one, such as choosing \( x \geq 6 \), the range remains unaffected as \([0, \infty)\) since all the values above 0 are still covered.

Graphical Analysis

Graphical analysis helps us visualize how a function behaves and how domain restrictions affect it. For the absolute value function \( f(x) = |x-6| \), graphing shows a symmetrical "V" shape with its vertex at \( x = 6 \).

By restricting the domain, say to \( x \geq 6 \), we reduce the graph to the right half of the "V". This side is monotonically increasing, ensuring a one-to-one relationship.

Graphical analysis is a powerful tool to confirm that both the desired domain restriction and range are maintained. It visually confirms that the one-to-one condition is met, validating our restrictions.