Problem 46 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 43-46: 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 \(y\). $$ y=2 \sin ^{-1}(3 x-4) $$

Step-by-Step Solution

Answer

Domain: \(1 \leq x \leq \frac{5}{3}\). Range: \(-\pi \leq y \leq \pi\). Solution for \(x\): \(x = \frac{\sin(\frac{y}{2}) + 4}{3}\).

Step 1: Determine the domain of the function

The function is presented as \( y = 2 \sin^{-1}(3x - 4) \). Recall that the domain of \( \sin^{-1}(u) \) is given by \(-1 \leq u \leq 1\). Here, \( u = 3x - 4 \), therefore the inequality becomes \(-1 \leq 3x - 4 \leq 1\). Solving this inequality:1. Add 4 to all parts: \(-1 + 4 \leq 3x \leq 1 + 4\)2. Simplify to get: \(3 \leq 3x \leq 5\)3. Divide the entire inequality by 3 to isolate \(x\): \[1 \leq x \leq \frac{5}{3}\]Therefore, the domain of \(f\) is \(1 \leq x \leq \frac{5}{3}\).

Step 2: Determine the range of the function

The range of the function \(y = 2 \sin^{-1}(3x - 4)\) depends on the range of the expression \(\sin^{-1}(u)\), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Because the given function has a coefficient of 2, the output of \(\sin^{-1}(3x - 4)\) is multiplied by 2.Thus, the range of \(2 \sin^{-1}(3x - 4)\) is:\[-\pi \leq y \leq \pi\]

Step 3: Solve for x in terms of y

We are given the equation \(y = 2 \sin^{-1}(3x - 4)\). First, we need to isolate the \(\sin^{-1}\) function:1. Divide both sides by 2: \(\frac{y}{2} = \sin^{-1}(3x - 4)\)2. Apply the sine function to both sides to cancel the inverse sine: \(\sin(\frac{y}{2}) = 3x - 4\)3. Solve for \(x\) by adding 4 to both sides: \(3x = \sin(\frac{y}{2}) + 4\) 4. Finally, divide by 3 to isolate \(x\): \[x = \frac{\sin(\frac{y}{2}) + 4}{3}\]

Key Concepts

Domain and Range

When we talk about the domain of a function, we're referring to all the possible input values that won't cause any troubles like division by zero or square roots of negative numbers. For the given equation, \(y = 2 \sin^{-1}(3x - 4)\), the concern is the inverse sine part, \(\sin^{-1}(u)\). The domain of \(\sin^{-1}(u)\) is confined to the interval \([-1, 1]\), meaning the expression inside must also remain within this interval.
To find the domain for \(f\), set \(u = 3x - 4\) and solve \(-1 \leq 3x - 4 \leq 1\). Adding 4 to each part simplifies the inequality to \(3 \leq 3x \leq 5\). Dividing everything by 3, we find that \(x\) must lie between 1 and \(\frac{5}{3}\).
This inequality gives us the domain of \(f\): \(1 \leq x \leq \frac{5}{3}\). It's critical because it tells us exactly which inputs are valid.

• The domain is limited by the condition \(-1 \leq 3x - 4 \leq 1\).
• Valid \(x\) values must fall between 1 and \(\frac{5}{3}\).

As for the range, this refers to all possible values the function \(f\) can output, determined by multiplying the range of \(\sin^{-1}(3x - 4)\), namely \([-\frac{\pi}{2}, \frac{\pi}{2}]\), by 2. Hence, the range of \(f\) is \([-\pi, \pi]\), representing the span of possible outputs.

Solving Trigonometric Equations

Solving trigonometric equations is like solving any other equation, but with the added step of dealing with trigonometric functions. Here, the task is to solve \(y = 2 \sin^{-1}(3x - 4)\) for \(x\) in terms of \(y\).
The equation involves the inverse sine function \(\sin^{-1}(u)\), which can be a bit tricky. To solve it, we start by isolating the inverse sine.

• First, divide both sides by 2, giving \(\frac{y}{2} = \sin^{-1}(3x - 4)\).
• To remove the inverse sine, apply the sine function to both sides, resulting in \(\sin(\frac{y}{2}) = 3x - 4\).
• Add 4 to both sides to shift the equation: \(3x = \sin(\frac{y}{2}) + 4\).
• Finally, divide by 3 to isolate \(x\): \[x = \frac{\sin(\frac{y}{2}) + 4}{3}\].

This approach reveals the relationship between \(x\) and \(y\), demonstrating how changes in \(y\) reflect in \(x\). Understanding this corresponds to interpreting how the curve looks around its domain.

Algebraic Manipulation

Algebraic manipulation is key to transforming equations so you can solve them effectively. When working with inverse trigonometric functions, such manipulation requires careful handling of trigonometric identities and properties.
For the given equation \(y = 2 \sin^{-1}(3x - 4)\), several steps of algebraic manipulation are needed to solve for \(x\) in terms of \(y\):

• Start by isolating the part involving the inverse sine: divide both sides by 2, obtaining \(\frac{y}{2} = \sin^{-1}(3x - 4)\).
• Remove the inverse function by taking the sine of both sides, shifting to \(\sin(\frac{y}{2}) = 3x - 4\).
• Rearrange the equation to solve for \(3x\), adjusting both sides to get \(3x = \sin(\frac{y}{2}) + 4\).
• Finally, divide by 3 to isolate \(x\), leading to the solution \(x = \frac{\sin(\frac{y}{2}) + 4}{3}\).

These steps highlight how algebraic manipulation helps unravel complicated relationships in equations, making them simpler and easier to understand.

Function Analysis

Function analysis involves understanding how a function behaves, its characteristics, and its implications for the inputs and outputs. For \(y = 2 \sin^{-1}(3x - 4)\), knowing the domain, range, and solving variable relations helps us decode this function.
Let's put these pieces together:

• The domain \(1 \leq x \leq \frac{5}{3}\) tells us the possible inputs.
• The range \([-\pi, \pi]\) lets us know possible outputs.
• Solving for \(x\) shows how \(y\) affects \(x\).

Combining these insights, we learn that as \(x\) varies within its domain, \(y\) covers a broad range. The function's structure \(2 \sin^{-1}(3x - 4)\) implies that not every real number is a valid output, limited by trigonometry. Function analysis doesn't merely focus on numbers but rather unveils deeper relationships and behaviors within functions.