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.

Determine the domain of definition of the following functions:

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

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) $g(x) = x^3 - 3x + 1$

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?

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

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:

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