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The tangent function, denoted as \(\tan(x)\), is one of the primary trigonometric functions. It relates an angle in a right triangle to the ratio of the opposite side to the adjacent side. For a given angle \(x\): \[\tan(x) = \frac{opposite}{adjacent}\]. This function is periodic, with a period of \(\frac{\beta_Pi}{2}\), which means it repeats its values every \(\frac{\beta_Pi}{2}\) radians. The tangent function has no horizontal symmetry and its graph has vertical asymptotes, where the function is undefined. These asymptotes occur at \(x = \frac{\beta_Pi}{2}, \frac{3\beta_Pi}{2}, \frac{5\beta_Pi}{2}, ...\).In addition to its graphical representation, \(\tan(x)\) has distinct properties:

• It is undefined for \(x = \frac{\beta_Pi}{2} + k\beta_Pi\), where \(k\) is an integer.
• The range of \(\tan(x)\) includes all real numbers.
• It has a periodicity of \(\frac{\beta_Pi}{2}\) radians, making the function repeat every \(\frac{\beta_Pi}{2}\) radians along the x-axis.

The arctangent function, denoted as \(\tan^{-1}(x)\) or \(\text{arctan}(x)\), is the inverse of the tangent function. Essentially, this function takes a value (which is a tangent) and returns the angle whose tangent is that value. For example, \(\tan^{-1}(-4.2)\) gives the angle \(\theta\) such that \( \tan(\theta) = -4.2 \).Some key properties of the arctangent function include:

• The range of \(\tan^{-1}(x)\) is limited to \(-\frac{\beta_Pi}{2} \leq \tan^{-1}(x) \leq \frac{\beta_Pi}{2}\).
• It is an odd function, meaning \(\tan^{-1}(-x) = -\tan^{-1}(x)\).
• Its graph is continuous and passes through the origin.

When solving problems with arctangent, it is essential to remember these properties, as they dictate how the function behaves and ensures the correct interpretation and solutions.

Inverse functions reverse the effect of the original function. If \(f(x)\) is a function, its inverse \(f^{-1}(x)\) will yield \(x\) when \(f(x)\) is applied. So, \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).For inverse trigonometric functions like \(\tan^{-1}(x)\), this property means:

• If \(y = \tan^{-1}(x)\), then \(\tan(y) = x\).
• This is the core idea behind the exercise: find the value of \(\tan^{-1}(-4.2)\) and then apply the \(\tan\) function.

Using the inverse property helps to simplify complex expressions and solve equations. In this problem, we set \( \theta = \tan^{-1}(-4.2)\), which means \( \tan(\theta) = -4.2 \). Hence, \( \tan(\tan^{-1}(-4.2)) = -4.2 \), reflecting how the original input is recovered through the inverse function.