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

Step-by-Step Solution

Answer

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

Step 1: Identify the function

The function given is \( y = 3 \tan^{-1}(2x + 1) \). Here, \( \tan^{-1} \) represents the inverse tangent function, also known as arctangent.

Step 2: Determine the Domain

The arctangent function has a domain of all real numbers, meaning \( 2x + 1 \) can be any real number. Consequently, the domain of \( f(x) \) is also all real numbers: \( x \in \mathbb{R} \).

Step 3: Determine the Range

The range of the \( \tan^{-1} \) function is \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \). Multiplying by 3 to obtain \( y = 3 \tan^{-1}(2x + 1) \), the range becomes \( \left( -\frac{3\pi}{2}, \frac{3\pi}{2} \right) \).

Step 4: Solve for x in terms of y

To express \( x \) in terms of \( y \), first divide by 3 to isolate the arctangent: \( \tan^{-1}(2x + 1) = \frac{y}{3} \). Then apply the tangent function: \( 2x + 1 = \tan\left(\frac{y}{3}\right) \). Finally, solve for \( x \): \( x = \frac{\tan\left(\frac{y}{3}\right) - 1}{2} \).

Key Concepts

Understanding Arctangent

One of the key components in the given equation is \( \tan^{-1} \), which represents the arctangent function. Arctangent is the inverse function of the tangent. This means that if \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \). It allows us to find an angle if we know the tangent value. The arctangent function is crucial in trigonometry because it helps in determining angles in right-angled triangles based on their tangent ratios. Arctangent specifically converts a real number back into its angle measure in radians. It is vital in calculus when solving integration and differentiation problems that involve trigonometric functions. A unique characteristic of arctangent is that inverting the tangent function modifies its domain and range, adapting it to a new purpose.

Domain and Range Explained

The domain of a function is essentially the set of all possible input values (\( x \) values) that the function can accept without causing any undefined or non-real result. For the arctangent function, the domain encompasses all real numbers. In simple terms, no matter what number you input, \( \tan^{-1} \) can find a corresponding angle for it.

The range of a function breaks down what potential output values (\( y \) values) we can obtain from the function. For \( \tan^{-1}(x) \), the range is restricted to \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), meaning it will only produce angle values within this interval. When multiplied by a scalar, like 3 in our equation, the range also stretches or compresses proportionally. Thus, the range of \( y = 3 \tan^{-1}(2x + 1) \) extends to \( \left(-\frac{3\pi}{2}, \frac{3\pi}{2}\right) \). Ensuring comprehension of domain and range facilitates understanding the behavior of functions across their scopes.

Solving Equations Involving Arctangent

When presented with an equation like \( y = 3 \tan^{-1}(2x + 1) \) and asked to solve for \( x \) in terms of \( y \), the process involves reversing the operations originally performed on \( x \). Begin by isolating the arctangent function. Divide \( y \) by 3 to reverse the multiplication, resulting in \( \tan^{-1}(2x + 1) = \frac{y}{3} \).

Next, apply the tangent function to both sides to cancel out the arctangent, transforming the equation into \( 2x + 1 = \tan\left(\frac{y}{3}\right) \). Finally, solve for \( x \) by isolating it on one side of the equation, yielding \( x = \frac{\tan\left(\frac{y}{3}\right) - 1}{2} \). Such steps in solving equations incorporate logical reverse operations and manipulation to determine the desired variable in terms of another, a nifty trick in algebraic and trigonometric problem-solving.