Problem 44 - 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 \(y\) $$y=3 \tan ^{-1}(2 x+1)$$

Step-by-Step Solution

Answer

Domain: \(\mathbb{R}\); Range: \((-\frac{3\pi}{2}, \frac{3\pi}{2})\); \(x = \frac{\tan(\frac{y}{3}) - 1}{2}\).

Step 1: Understanding the Function Structure

The given function is \( y = 3 \tan^{-1}(2x + 1) \), where \( \tan^{-1}(x) \) is the inverse tangent function, also known as arctan. Trigonometric inverse functions are common in calculating angles given a ratio of sides in a right triangle.

Step 2: Finding the Domain of the Function

The principal domain of \( \tan^{-1}(x) \) is all real numbers, since arctan can accept any real number input and produce an angle as output. Therefore, since \( \tan^{-1}(2x+1) \) can accept any value of \( x \), the domain is all real numbers: \( \mathbb{R} \).

Step 3: Calculating the Range of the Function

The range of \( \tan^{-1}(x) \) is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Multiplying by 3, as in \( 3 \tan^{-1}(2x + 1) \), scales the range by 3, giving \( \left(-\frac{3\pi}{2}, \frac{3\pi}{2}\right) \). Hence, the range of \( f(x) \) is \( \left(-\frac{3\pi}{2}, \frac{3\pi}{2}\right) \).

Step 4: Solving for x in terms of y

Start by isolating the \( \tan^{-1} \) function by dividing the equation by 3: \( \tan^{-1}(2x+1) = \frac{y}{3} \). Apply the tangent function to both sides, inverting the arctan: \( 2x+1 = \tan\left(\frac{y}{3}\right) \). Finally, solve for \( x \): \( x = \frac{\tan\left(\frac{y}{3}\right) - 1}{2} \).

Key Concepts

Domain of a Function

The domain of a function is all the possible input values (usually denoted as 'x') that will produce a valid output. For the function given by:
\( y = 3 \tan^{-1}(2x + 1) \), our goal is to find all possible values of \( x \) that ensure the expression inside the inverse tangent, \( \tan^{-1}(2x + 1) \), makes sense.
The inverse tangent function, \( \tan^{-1}(x) \), can accept any real number, meaning it doesn't have restrictions like some other trigonometric inverse functions.
Thus, unlike functions that might have restrictions (such as fractions that must not have zero denominators or square roots that require non-negative numbers), the domain here includes all real numbers. That is:

• The domain is the set of all real numbers, \( \mathbb{R} \).
• This means you can input any real number into \( 3 \tan^{-1}(2x + 1) \) and get a corresponding output value for \( y \).

Range of a Function

The range of a function refers to all possible output values (usually denoted as 'y') that the function can produce. When considering the function \( y = 3 \tan^{-1}(2x + 1) \), the range is determined by how the inverse tangent function behaves.
For \( \tan^{-1}(x) \), the range is \((-\frac{\pi}{2}, \frac{\pi}{2})\), corresponding to the angles the inverse tangent can output.
Since our function involves multiplying \( \tan^{-1}(2x + 1) \) by 3, it stretches the angle results by a factor of 3. This operation expands the range to:

• From \(-\frac{\pi}{2} \times 3\) to \(\frac{\pi}{2} \times 3\).
• Mathematically, the range becomes \(\left(-\frac{3\pi}{2}, \frac{3\pi}{2}\right)\).
• Thus, for any real number \( x \), the function will produce a \( y \) value within this interval.

Solving Equations

Solving equations generally involves finding all the possible values of the unknown variable that satisfy the equation. In this exercise, we want to find \( x \) in terms of \( y \) from the given expression:
\( y = 3 \tan^{-1}(2x + 1) \).
To approach this, we begin by isolating the inverse trigonometric expression, setting up steps to 'undo' the operations applied to \( x \):

• Divide both sides by 3 to focus on the inverse tangent part: \( \tan^{-1}(2x+1) = \frac{y}{3} \).
• Apply the tangent function to both sides to reverse the inverse tangent: \( 2x + 1 = \tan\left(\frac{y}{3}\right) \).
• Isolate \( x \): subtract 1 from both sides, and divide by 2. The solution is \( x = \frac{\tan\left(\frac{y}{3}\right) - 1}{2} \).

This process takes advantage of mathematical operations that reverse each other, allowing us to solve for unknowns even in complex expressions involving trigonometric functions.