Table and graph of a function

Given the linear function $f(x) = 2x - 3$:

1. a) Complete a table of values for $x \in \{-2, 0, 1, 3\}$.
2. b) Draw the graph of the function in the Cartesian plane using the points obtained.

You can do it on paper and send a photo with your answer.

Consider the quadratic function $f(x) = x^2 - 4$:

1. a) Calculate the images for $x = -3, -2, 0, 2, 3$ and organise the results in a table.
2. b) Draw the graph of the function and identify the coordinates of its vertex.

You can do it on paper and send a photo with your answer.

Given the radical function $f(x) = \sqrt{x+2}$:

1. a) Determine its domain analytically.
2. b) Complete a table of values for $x \in \{-2, -1, 2, 7\}$ and use it to sketch the graph.

Given the rational function $f(x) = \frac{4}{x-2}$:

1. a) Determine its domain.
2. b) Construct a table of values evaluating the function at points close to $x = 2$ from the left and from the right ($x = 1,9; 1,99; 2,1; 2,01$).
3. c) Based on the results obtained, what graphical feature does the function display at $x = 2$?

You can do it on paper and send a photo with your answer.

Given the piecewise function:

$f(x) = \begin{cases} -x + 2 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \ge 1 \end{cases}$

1. a) Construct a table of values by evaluating the function at $x = -2; 0; 0,99; 1; 2; 3$.
2. b) Draw the graph of the function and state whether it is continuous at $x = 1$.

You can do it on paper and send a photo with your answer.

Given the piecewise function depending on a parameter $k$:

$f(x) = \begin{cases} x^2 - k & \text{if } x < 2 \\ 2x + 1 & \text{if } x \ge 2 \end{cases}$

1. a) Find the value of $k$ so that the two branches of the function "connect" at the point $x = 2$ (that is, so that the function is continuous).
2. b) Using the value of $k$ found, construct a table of values in the interval $[0, 4]$ and sketch the graph of the function.

You can do it on paper and send a photo with your answer.

The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of time $t$ (in minutes) since it is served.

From the graph:

1. a) State the initial temperature of the coffee.
2. b) What value does the temperature approach as a large amount of time elapses? What physical meaning does the horizontal asymptote have?
3. c) Is the function increasing or decreasing? Justify your answer.

Observe the following graph of a rational function of the form $f(x) = \frac{ax+b}{cx+d}$:

From the information provided by the graph (asymptotes and axis intercepts):

1. a) Determine the domain and range of the function.
2. b) Deduce the algebraic expression of the function $f(x)$.