Exercises: 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) Plot the function on the Cartesian plane using the points obtained.

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

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

Observe the following graph of a function $f(x)$ defined by straight line segments:

From the graph, determine:

1. a) The domain and the range of the function.
2. b) The value of $f(-2)$ and $f(1)$.

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.

Find the x- and y-intercepts of the rational function $f(x) = \frac{3x+6}{x-1}$ analytically.

Given the absolute value function $f(x) = |2x - 5|$:

1. a) Evaluate the function at the points $x = 0, 2, 2.5, 3, 5$ and construct a table of values.
2. b) Plot the function.

Identify whether the following table of values corresponds to a linear function or a quadratic function by calculating the differences between consecutive values of $y$. Then, find its analytical expression $y = f(x)$.

The cost $C(t)$ in dollars of leaving a car in a parking lot is given by the function $C(t) = 2 + 1.5t$, where $t$ is the time in hours.

1. a) Make a table of values for the first 4 hours ($t=0, 1, 2, 3, 4$).
2. b) Plot the function in that interval.

The temperature $T$ (in $^\circ\text{C}$) of an industrial oven since it is turned on is given by the function $T(t) = 20 + 15t$, where $t$ is measured in minutes.

1. a) What is the initial temperature of the oven?
2. b) Construct a table of values for $t = 0, 5, 10, 15$.
3. c) How much time must elapse for the oven to reach $200^\circ\text{C}$?

Given the piecewise-defined 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) Plot the function and indicate whether it is continuous at $x = 1$.

Consider the quadratic function $f(x) = -x^2 + 6x - 5$.

1. a) Calculate analytically the coordinates of its vertex and the x- and y-intercepts.
2. b) Create a table of values taking two points to the left of the vertex and two to the right.
3. c) Plot the parabola.

The following graph shows the evolution of the temperature $T$ (in $^\circ\text{C}$) of a cup of coffee as a function of the 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 tend towards as a lot of time passes? What is the physical meaning of the horizontal asymptote?
3. c) Is the function increasing or decreasing? Justify your answer.

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

1. a) Determine its domain.
2. b) Construct a table of values by 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) From the results obtained, what graphical feature does the function have at $x = 2$?

In a laboratory, the growth of a population of bacteria is studied. The number of bacteria $N(t)$ in thousands is given by the function $N(t) = 100 \cdot 2^t$, where $t$ is the time in hours.

1. a) Construct a table of values for the first 4 hours ($t = 0, 1, 2, 3, 4$).
2. b) Plot the function. What type of growth is observed?

Consider the function $f(x) = \sqrt{9-x^2}$.

1. a) Calculate its domain analytically.
2. b) Make a table of values for $x \in \{-3, -2, 0, 2, 3\}$.
3. c) Plot the function and state its range.

Given the absolute value function $f(x) = |x^2 - 4|$:

1. a) Express the piecewise-defined function by removing the absolute value bars.
2. b) Find the x- and y-intercepts.
3. c) Construct a table of values in the interval $[-3, 3]$ and draw its graph.

The monthly profit $B(x)$, in dollars, of an electronic components factory as a function of the number of units produced $x$ is modeled by the function $B(x) = -5x^2 + 300x - 4000$.

1. a) Find the break-even points (production for which the profit is zero).
2. b) Generate a strategic table of values that includes the break-even points and the vertex.
3. c) How many units must be produced to maximize the profit and what is this maximum profit?

A quadratic function of the form $f(x) = ax^2 + bx + c$ passes through the points $(0, 3)$, $(1, 0)$, and $(2, -1)$.

1. a) Set up and solve the system of equations to find the values of $a, b$ and $c$.
2. b) With the analytical expression obtained, calculate the vertex and construct a table of values to plot the parabola.

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) With the value of $k$ found, construct a table of values in the interval $[0, 4]$ and sketch the graph of the function.

The mass $M(t)$ in ounces of a radioactive isotope decreases with time $t$ (in days) according to the exponential function $M(t) = M_0 \cdot (0.5)^{\frac{t}{T}}$, where $M_0$ is the initial mass and $T$ is the half-life.

Knowing that at time $t=0$ there were $200\text{ oz}$ of substance and that after $60\text{ days}$ there are $50\text{ oz}$ left:

1. a) Calculate the value of the parameters $M_0$ and $T$.
2. b) Construct a table of values for $t = 0, 30, 60, 90, 120$ and sketch the graph.

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 intercepts):

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

Given the logarithmic function $f(x) = \log_2(x+4)$:

1. a) Determine analytically its domain and the equation of its vertical asymptote.
2. b) Construct a table of values by choosing strategic values of $x$ that result in exact powers of base 2 for the argument of the logarithm.
3. c) Plot the graph of the function.

We want to build a rectangular pen using an existing wall for one of its sides. There are $20\text{ m}$ of metal fencing available for the other three sides.

1. a) Express the area $A(x)$ of the pen as a function of the length $x$ of one of the sides perpendicular to the wall.
2. b) Determine the domain of validity of the function $A(x)$ in the context of the problem.
3. c) Construct a table of values and use it to sketch the graph and find the dimensions that maximize the area.

The following figure shows the graph of the exponential function $f(x) = 2^x$ and the line $y = x$.

1. a) From the highlighted points on the blue curve of $f(x)$, construct a table of values for its inverse function $f^{-1}(x)$.
2. b) Draw the graph of the inverse function in your notebook, taking advantage of the symmetry with respect to the line $y=x$.
3. c) Write the analytical expression of $f^{-1}(x)$.