#### Equation arctan

Consider $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$. The solution of the equation $f\left(\dfrac{1}{2}-x \right) + \arctan\left( \dfrac{\sqrt{3}}{3}\right)=\dfrac{\pi}{6}$ is:

# Complements of differential calculus in real numbers

Training
13

#### Inverse arctan

Consider $f(x)= a -2 \arctan(1-2x), \ a \in \mathbb{R}$, with domain $D_f=\mathbb{R}$. Determine the analytic expression of the inverse function of $f$, $f^{-1}(x)$, and use that to calculate the value of $a \in \mathbb R$ knowing that $f^{-1}\left( -\dfrac{\pi}{2}\right) =0$.

# Complements of differential calculus in real numbers

Training
13

#### Domain arctan

The domain ($D_f$) and the range ($D'_f$) of the function $f(x)=- 2 \left|\dfrac{\pi}{2} -2 \arctan(1-2x)\right|$ are, respectively,

# Complements of differential calculus in real numbers

Training
13

#### Domain arctan

The domain ($D_f$) and the range ($D'_f$) of the function $f(x)= -2 \arctan\left(2x\right)$ are, respectively,

# Complements of differential calculus in real numbers

Training
13

#### Domain arctan

The domain ($D_f$) and the range ($D'_f$) of the function $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ are, respectively,

# Complements of differential calculus in real numbers

Training
13

#### Sum of powers

What is the final value of the following sum: $2^4+3^0+6^2$

# Powers

Training
8