Basic concepts

Given the polynomial function $f(x) = 2x^2 - 5x + 1$, calculate the images for the values $x = -2$, $x = 0$ and $x = 3$.

A bicycle hire service charges a fixed fee of $3$ euros to register for the service, plus $1.50$ euros per hour of use. Write the algebraic expression of the function that relates the total cost to the time of use, and identify which is the independent variable and which is the dependent variable.

A taxi company charges $3$ euros standing charge and $1.20$ euros per kilometre travelled.

1. a) Write the function relating the total price of the journey to the kilometres travelled.
2. b) What is the domain of this function taking into account the context of the problem?
3. c) How much will a journey of $15$ km cost?
4. d) If a customer paid $21$ euros, how many kilometres did they travel?

Determine the domain of definition of the following functions:

1. a) $f(x) = \frac{3}{x - 5}$
2. b) $g(x) = \sqrt{x + 2}$

From the graph of the function $h(x)$, knowing that the $y$-coordinate of the open point at $x = -2$ is $1.25$, state its domain and range in interval notation.

Find the domain of definition of the function $f(x) = \frac{\sqrt{x + 4}}{x - 2}$.

Determine analytically the domain of definition of the function $f(x) = \sqrt{\frac{x - 3}{x + 2}}$. Express the result as a union of intervals.

Calculate analytically the points of intersection with the x-axis and the y-axis of the function $f(x) = x^2 - 6x + 8$.

Determine algebraically whether the following functions are even, odd, or have no symmetry:

1. a) $f(x) = x^4 - 2x^2$
2. b) $g(x) = x^3 - 3x + 1$

When studying the function $f(x) = \dfrac{2x - 6}{x + 1}$, Hugo makes a very common conceptual error when finding the intercepts with the axes. In his notebook, he writes:

"To find the intercept with the $Y$-axis, I solve the equation $2x - 6 = 0$, from which I get $x = 3$, so the graph intersects the $Y$-axis at the point $(3, 0)$. To find the intercept with the $X$-axis, I substitute $x = 0$ and calculate $f(0) = \dfrac{-6}{1} = -6$, so it intersects the $X$-axis at the point $(0, -6)$".

1. a) Explain in detail what confusion Hugo had regarding the roles of the independent and dependent variables when finding the intercepts with the axes.
2. b) Reason, from the definition of a function, why it is mathematically impossible for the graph of a function to intersect the $Y$-axis at two or more distinct points.
3. c) Correct Hugo's procedure and determine the correct coordinates of the intersection points of $f(x)$ with both Cartesian axes.

Calculate the average rate of change of the function $f(x) = x^2 - 4$ over the intervals $[-1, 2]$ and $[2, 5]$.

The distance $d$ (in metres) covered by a free-falling object as a function of time $t$ (in seconds) is approximated by the function $d(t) = 5t^2$. Calculate the average speed of the object (which corresponds to the average rate of change) between the first and third seconds ($t \in [1, 3]$), and between the third and fifth seconds ($t \in [3, 5]$).

Calculate the average rate of change of the function $f(x) = \frac{1}{x}$ over the intervals $[1, 2]$ and $[2, 4]$. What does the sign of the results indicate about the monotonicity of the function over those intervals?

Observe the following graph of a function $f(x)$ and determine its intervals of increase and decrease, as well as the exact coordinates of its local maximums and minimums.

The temperature $T$ (in degrees Celsius) inside an industrial oven from when it is turned on until it is turned off is given by the function $T(t) = -t^2 + 8t + 20$, where $t$ is the elapsed time in hours. At what moment is the maximum temperature reached? What is the value of this maximum temperature?

A farmer has $100$ metres of fencing to enclose a rectangular vegetable patch using the straight wall of an existing building, so that he only needs to fence three of the four sides of the rectangle.

1. a) Find the expression of the function that gives the area of the patch as a function of the length $x$ of one of the sides perpendicular to the wall.
2. b) Determine the valid domain of definition for $x$ in the context of this problem.
3. c) What dimensions should the patch have so that its area is maximum?

Determine analytically the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{2x + 1}{x - 3}$.

Design the algebraic expression of a rational function of the form $f(x) = \dfrac{a \times x + b}{x + c}$ that simultaneously satisfies the following three conditions:

1) Its domain of definition is $\mathbb{R} \setminus \{2\}$.

2) Its graph intersects the axis of ordinates ($Y$-axis) at the point $(0, -3)$.

3) Its graph intersects the axis of abscissas ($X$-axis) at the point $(3, 0)$.

Explain step by step how you deduce the numerical value of each of the parameters $a$, $b$, and $c$, and write the final formula of the function, verifying that the numerator does not vanish at the value excluded from the domain.

Given the rational function $f(x) = \frac{ax^2 + 3}{2x^2 - x}$, determine the value of the parameter $a$ so that the function has a horizontal asymptote at the line $y = 4$. With this value of $a$ calculated, also find the equations of the vertical asymptotes of the function.

Given the following graph of a piecewise-defined function, find the values of $f(-2)$, $f(1)$ and $f(4)$. Then, state at which points the function is discontinuous and the type of each discontinuity.

The accompanying graph represents the height in metres of a Ferris wheel capsule as a function of time in minutes. Observe the graph and answer:

1. a) Is it a periodic function? If so, state its period.
2. b) What are the maximum and minimum heights reached by the capsule?
3. c) At what height is the capsule exactly $10$ minutes after starting to rotate?

Find the values of the parameters $a$ and $b$ so that the following piecewise-defined function is continuous throughout its domain:

$f(x) = \begin{cases} 2x + a & \text{if } x < 1 \\ x^2 - 3x + 5 & \text{if } 1 \le x < 4 \\ bx - 1 & \text{if } x \ge 4 \end{cases}$