3. Dadas las funciones \( f: D_{f} \rightarrow \mathbb{R}, f(x)=\log _{2}\left(\frac{x+1}{4}\right) \) y \( g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=4 x^{2}-1 \). [25 puntos \( ] \) (a) Determinar el dominio \( D_{f} \), y definir la función inversa de \( f \). (b) En un mismo sistema de coordenadas cartesianas, realizar las gráficas de las funciones \( f^{-1} \) y \( f \), identificando intersección con los ejes coordenados y asintotas. (c) Definir las funciones \( f \circ g \) y \( \frac{f}{g} \).

QUESTION 7 Given the function with the equation \( k(x)=\frac{-6}{x+2}-2 \) : \( 7.1 \quad \) Write the equations of the asymptotes of \( k \).

Clearly indicate all intercepts with the axes as well as the QUESTION 7 Given the function with the equation \( k(x)=\frac{-6}{x+2}-2 \) : 7.1 Write the equations of the asymptotes of \( k \). 7.2 Give the range of \( k \). 7.3 Give the domain of \( k \).

If \( (-1,8) \) is a point on the graph of \( y=f(x) \), which of the following must be on the graph of \( y=f(-x) \) ? \( (8,1) \) \( (-1,-8) \) Choose the correct answer below. \( (1,8) \) \( (-8,-1) \)

Find the function that is finally graphed after the following transformations are applied to the graph of \( y=\sqrt{x} \) in the order listed. (1) Shift up 7 units (2) Reflect about the \( x \)-axis (3) Reflect about the \( y \)-axis \( y= \)

(a) Express the distance \( d \) from \( P \) to the origin as a function of \( x \). \( d(x)=\sqrt{\mathrm{x}^{4}-3 \mathrm{x}^{2}+4} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (b) What is \( d \) if \( x=0 \) ? If \( x=0 \), the distance \( d \) is (Type an integer or decimal rounded to two decimal places as needed.)