#### Inverse arctan

Consider $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$ and range $D'_f = \left] -\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right[$. The analytic expression of the inverse function of $f$, $f^{-1}(x)$, its domain ($D_{f^{-1}}$) and range ($D'_{f^{-1}}$) are, respectively,

# Complements of differential calculus in real numbers

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#### Domain arccos

Let $f(x)= a +2 \arccos(3x+b)$, with $a, b \in \mathbb R$. Knowing that the domain and range of $f$ are, respectively, $D_f=\left[ -\dfrac{2}{3}, 0\right]$ and $D'_f=\left[ -3\pi, -\pi\right]$, then $a$ and $b$ are:

# Complements of differential calculus in real numbers

Training
13

#### Domain arccos

Consider the functions $f(x)= 5 \arccos\left(2x\right)$ with domain $D_f$ and range $D'_f$ and $g(x)= \arccos\left(\dfrac{x}{2}\right)-3\pi$ with domain $D_g$ and range $D'_g$. State whether the following statements are true (T) or false (F).

# Complements of differential calculus in real numbers

Training
13

#### Domain arccos

The domain ($D_f$) of the function $f(x)= -3\pi + 2\arccos\left( \dfrac{1}{3x+1}\right)$ is:

# Complements of differential calculus in real numbers

Training
13

#### Inverse arccos

Consider $f(x)= -3\pi +2 \arccos(3x+1)$ with domain $D_f=\left[ -\dfrac{2}{3}, 0\right]$ and range $D'_f = \left[ -3\pi, -\pi\right]$. The analytic expression of the inverse function of $f$, $f^{-1}(x)$, its domain ($D_{f^{-1}}$) and $f^{-1}(-\pi)$ are, respectively,

# Complements of differential calculus in real numbers

Training
13

#### Inverse arccos

Consider the functions $f(x)= 5 \arccos\left(2x\right)$ with domain $D_f = \left[-\dfrac{1}{2}, \dfrac{1}{2}\right]$ and range $D'_f = \left[0, 5\pi\right]$ and $g(x)= \arccos\left(\dfrac{x}{2}\right)-3\pi$ with domain $D_g = \left[-2, 2\right]$ and range $D'_g = \left[-3\pi, -2\pi\right]$. State whether the following statements are true (T) or false (F).

# Complements of differential calculus in real numbers

Training
13