Questions: Find the exact value of the composite function without using a calculator. [ tan (tan ^-1 3)= ] Note: Enter an exact simplified answer with no decimals. Use pi for π if necessary. Enter none if there is no answer.

Solution Steps

To find the exact value of the composite function \(\tan \left(\tan^{-1} 3\right)\), we need to understand the relationship between the tangent function and its inverse. The inverse tangent function, \(\tan^{-1}(x)\), gives us an angle whose tangent is \(x\). Therefore, \(\tan(\tan^{-1}(3))\) should simply return the value 3.

We need to evaluate the expression \(\tan \left(\tan^{-1} 3\right)\). The function \(\tan^{-1}(x)\) returns an angle \(\theta\) such that \(\tan(\theta) = x\). In this case, we have \(\theta = \tan^{-1}(3)\).

Since \(\theta = \tan^{-1}(3)\), it follows that: \[ \tan(\theta) = 3 \] Thus, substituting back into our original expression, we find: \[ \tan \left(\tan^{-1} 3\right) = 3 \]

Final Answer