Problem 74 - 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. $$-2 f(-x)$$

Step-by-Step Solution

Answer

Domain: \([-2, 1]\), Range: \([-6, 0]\)

Step 1: Understand the transformation of the domain

The original function \( f(x) \) has a domain of \([-1, 2]\). When we substitute \(-x\) into the function, the domain transforms to account for the negative input of \(x\). This means the expression \( -x \) is now bounded by the reverse of the original domain limits.Specifically, consider the inequalities:\[-2 \leq -x \leq 1\]This can be solved as:1. Multiply each part by \(-1\) (note this reverses the inequality signs):\[2 \geq x \geq -1\]Which shows the domain remains \([-2, 1]\) but flipped.

Step 2: Apply the transformation to find the range

The original function \( f(x) \) has a range of \([0, 3]\). The transformation \(-2f(x)\) will affect this range.1. Multiply the entire range of \( f(x) \) by \(-2\): - Multiply 0 by \(-2\): 0 \( \times \) \(-2\) = 0 - Multiply 3 by \(-2\): 3 \( \times \) \(-2\) = \(-6\)2. Thus, the new range will be \([-6, 0]\).

Step 3: Express the solution

Now that the transformations are applied, we conclude that the domain and range of \(-2 f(-x)\) are:- **Domain:** \([-2, 1]\)- **Range:** \([-6, 0]\)

Key Concepts

Domain and Range

Understanding the domain and range of a function is crucial to grasp how a function behaves over specific values of input and output.
The **domain** is essentially all the possible input values (generally considered as 'x-values') that a function can accept.In this problem, the original function has a domain of \([-1, 2]\). For the transformation described, with \(-x\), the domain becomes \([-2, 1]\). This is because when a variable is negated (as in \(-x\)), the direction of inequality signs is reversed, reflecting the domain over the y-axis.The **range**, on the other hand, refers to the output values (typically 'y-values') a function can produce. Originally, the range is \([0, 3]\), and after applying the transformation \(-2f(x)\), it morphs into \([-6, 0]\). This is achieved by multiplying each endpoint of the original range by \(-2\), which not only scales but also reverses the sign of each range limit, essentially reflecting it over the x-axis.

Function Notation

Function notation is a way to express a function using symbols, usually denoting a formula that describes a relationship between two quantities.Functions are usually denoted by letters such as \(f\), \(g\), or \(h\). In the exercise, we have \(f(x)\) representing a function of \(x\). The expression \(-2f(-x)\) provides specific details about how inputs are transformed before being processed by the function.It is important to understand that this notation helps to specify not just the function itself but also how it handles transformations:

• The negative sign before \(x\) indicates reflection over the y-axis.
• The multiplication by -2 applies both inversion and scaling vertically.

Using proper function notation makes it easier to comprehend and communicate the ways functions are manipulated or transformed.

Inequalities in Algebra

Inequalities in algebra help us understand the boundaries within which a function operates. They are used extensively to define domains, solve problems, and find constraints in algebraic expressions.In the exercise, multiplying the inequalities by \(-1\) is a key step. Remember that when multiplying or dividing both sides of an inequality by a negative number, the inequality sign switches direction. This is a fundamental principle that often causes confusion.

• For example: \(-2 \leq -x \leq 1\) becomes \(2 \geq x \geq -1\).
• Here, the directions of the inequalities are flipped to reflect the correct domain.

Paying close attention to these operations ensures accuracy in defining the constraints or limits for the domain and range of the transformed function.