Problem 72 - 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. $$-2 f(4 x)$$

Step-by-Step Solution

Answer

Domain: \([-\frac{1}{4}, \frac{1}{2}]\), Range: \([-6, 0]\).

Step 1: Determine the domain of the transformation

The function given is \[-2 f(4x)\]. To find the domain of \(f(4x)\), we need to solve for \(x\) in the domain of \(f(x)\) which is \([-1, 2]\).***Start by setting the expression inside of \(f\), i.e., \(4x\), equal to the domain of \(f(x)\):\[\begin{align*}-1 &\leq 4x \leq 2.\end{align*}\]***Divide each part of the inequality by 4:\[\begin{align*}-\frac{1}{4} &\leq x \leq \frac{1}{2}.\end{align*}\]Thus, the domain of \(f(4x)\) is \([-\frac{1}{4}, \frac{1}{2}]\).

Step 2: Determine the range of the transformation

Now consider the function \(-2 f(4x)\).The range of \(f(x)\) is given as \([0, 3]\).To find the range of \(-2 f(4x)\), apply the transformation to the range of \(f(x)\):- Multiply the entire range by -2:\[0 \times (-2) = 0 \quad \text{and} \quad 3 \times (-2) = -6.\]- Since multiplying by a negative number reverses the inequality, the range should be ordered:Thus, the range of \(-2 f(4x)\) is \([-6, 0]\).

Key Concepts

Domain and Range

When discussing functions, the domain refers to all possible input values, while the range represents all possible outputs that the function can produce. In the given exercise, the domain of the original function \(f(x)\) is \([-1, 2]\) and the range is \([0, 3]\). This means any \(x\) fed into \(f(x)\) must lie within \([-1, 2]\), and the output of \(f(x)\) will be within \([0, 3]\).

This particular problem involves a transformation function \(-2 f(4x)\). To determine the new domain, consider the transformation affect on \(x\); for \(f(4x)\), replace \(x\) with \(4x\) and solve the inequality \([-1 \leq 4x \leq 2]\).

• Divide each part of the inequality by 4, reaching \([-\frac{1}{4}, \frac{1}{2}]\).

Thus, after modifying \(x\) in the expression, the domain of \(f(4x)\) becomes \([-\frac{1}{4}, \frac{1}{2}]\).

Similarly, calculating the range requires consideration of the multiplication factor, which is \(-2\) here. The output of the function is affected directly by this factor, leading to the new range \([-6, 0]\).

Inequalities

Inequalities are crucial in determining domains when functions are transformed. They describe the range within which values of variables must lie. In the exercise, you deal with the inequality \(-1 \leq 4x \leq 2\) which comes from equating \(4x\) with the domain of the original function \([-1, 2]\).

Solving this inequality is straightforward:

• Divide the entire inequality by 4, affecting all parts: \(-\frac{1}{4} \leq x \leq \frac{1}{2}\).

Through this manipulation, you discover that \(x\) must lie within \([-\frac{1}{4}, \frac{1}{2}]\) for the transformed function \(f(4x)\).

When multiplying by a negative number, inequalities reverse. Multiplying the range \([0, 3]\) by \(-2\), to get \(-6\), notice the range order reverses: \([-6, 0] \). This reversing is a characteristic feature of inequalities when scaled by negative values.

Algebraic Manipulation

Algebraic manipulation involves transforming expressions or equations to simplify or find solutions. In the context of function transformations like \(-2 f(4x)\), manipulation helps derive the domain and range. You first focus on the expression within the function, \(4x\), aligning it with the original domain constraint \([-1, 2]\).

• By solving the inequality \(-1 \leq 4x \leq 2\), through division by 4, we get \([-\frac{1}{4}, \frac{1}{2}]\)

This approach simplifies the domain comparison post-manipulation.

The multiplication by \(-2\) in the function affects the output range directly. Each element in the range is multiplied separately by \(-2\) to yield \([0 \times (-2), 3 \times (-2)]\), simplifying to \([-6, 0]\). Algebraic manipulation here includes multiplying each endpoint and understanding how transformations (like negations) affect range ordering.