Problem 45 - Precalculus: Functions and Graphs

Question

The given equation has the form \(y=f(x)\) (a) Find the domain of \(f .\) (b) Find the range of \(f .\) (c) Solve for \(x\) in terms of \(y\) $$y=4 \cos ^{-1} \frac{2}{3} x$$

Step-by-Step Solution

Answer

(a) Domain: \([-\frac{3}{2}, \frac{3}{2}]\). (b) Range: \([0, 4\pi]\). (c) Solution: \(x = \frac{3}{2} \cos\left(\frac{y}{4}\right)\).

Step 1: Find the Domain of the Function

The given function is \( y = 4 \cos^{-1}\left(\frac{2}{3}x\right) \). The argument of the \( \cos^{-1} \) or inverse cosine function, \( \frac{2}{3}x \), must be in the interval \([-1, 1]\) to ensure that the function is defined and real. This gives the condition:\[-1 \leq \frac{2}{3}x \leq 1\]We need to solve this inequality for \( x \). Begin by multiplying through by \( \frac{3}{2} \) to eliminate the fraction:\[ -\frac{3}{2} \leq x \leq \frac{3}{2} \]Thus, the domain of \( f(x) \) is \([-\frac{3}{2}, \frac{3}{2}]\).

Step 2: Find the Range of the Function

The range of the \( \cos^{-1} \) function is \([0, \pi]\). Multiplying by 4 scales this range, so we will multiply the endpoints by 4:\[ 4(0) = 0 \]\[ 4(\pi) = 4\pi \]Thus, the range of the function \( f(x) \) is \([0, 4\pi]\).

Step 3: Solve for x in Terms of y

Rearrange the equation \( y = 4 \cos^{-1}\left(\frac{2}{3}x\right) \) to solve for \( x \).1. Divide both sides by 4 to isolate the \( \cos^{-1} \): \[ \cos^{-1}\left(\frac{2}{3}x\right) = \frac{y}{4} \]2. Apply cosine to both sides to eliminate the inverse cosine: \[ \frac{2}{3}x = \cos\left(\frac{y}{4}\right) \]3. Solve for \( x \) by multiplying by \( \frac{3}{2} \) to clear the fraction: \[ x = \frac{3}{2} \cos\left(\frac{y}{4}\right) \]

Key Concepts

Domain of a Function

The domain of a function in mathematics represents the set of all possible input values (x-values) that the function can accept without causing any mathematical exceptions, such as division by zero.
For inverse trigonometric functions like the inverse cosine (\(\cos^{-1}\)), the argument must adhere to specific constraints to ensure the function returns a real number.

• In the case of \(y = 4 \cos^{-1}\left(\frac{2}{3}x\right)\), the argument \(\frac{2}{3}x\) must be in the interval \([-1, 1]\).
• To find the domain, solve the inequality \(-1 \leq \frac{2}{3}x \leq 1\) for \(x\).
• Multiply each part of the inequality by \(\frac{3}{2}\) to clear the fraction, yielding \(-\frac{3}{2} \leq x \leq \frac{3}{2}\).

Thus, the domain of the function is \([-\frac{3}{2}, \frac{3}{2}]\), meaning \(x\) can take any value between \(-\frac{3}{2}\) and \(\frac{3}{2}\) inclusive.

Range of a Function

While the domain tells us about the possible inputs, the range informs us about the potential outputs or y-values of the function.
The range of a function depends highly on the type of function and any transformations applied to it.

• For the inverse cosine function \(\cos^{-1}\), the intrinsic range is \([0, \pi]\).
• In the equation \(y = 4 \cos^{-1}\left(\frac{2}{3}x\right)\), multiplying \(\cos^{-1}\) by 4 scales this range.
• Therefore, each endpoint of the interval \([0, \pi]\) needs to be multiplied by 4.
• This results in a new range of \([0, 4\pi]\).

Consequently, the function's range covers all possible outputs from 0 to \(4\pi\). So no matter what value within the domain \(x\) takes, \(y\) will always fall between 0 and \(4\pi\). This scaling is crucial in understanding how functions adapt and expand their range following a multiplication.

Solving Equations Involving Inverse Functions

Inverse trigonometric functions often require particular steps to isolate and solve for a variable. These solve processes are essential in uncovering the relationships among different elements in an equation.
To solve an equation like \(y = 4 \cos^{-1}\left(\frac{2}{3}x\right)\) for \(x\), follow these straightforward steps:

• First, isolate the inverse trigonometric function by dividing both sides of the equation by 4. Thus, \(\cos^{-1}\left(\frac{2}{3}x\right) = \frac{y}{4}\).
• Next, remove the inverse cosine by taking the cosine of each side: \(\frac{2}{3}x = \cos\left(\frac{y}{4}\right)\).
• Finally, solve for \(x\) by multiplying each side by \(\frac{3}{2}\) to resolve the fraction: \(x = \frac{3}{2} \cos\left(\frac{y}{4}\right)\).

These steps show how inverse trigonometric functions can be strategically managed to express an equation in terms of the desired variable. Understanding each transformation can help demystify complex functions and relationships.