Exercises: 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 rental service charges a fixed fee of $60$ pesos to register for the service, plus $30$ pesos 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.

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

Determine the domain of definition of the following functions:

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

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.

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

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

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.

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

The distance $d$ (in meters) 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 ($t \in [3, 5]$).

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

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.

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

A taxi company charges a base fare of $60$ pesos and $24$ pesos per kilometer traveled.

1. a) Write the function relating the total price of the trip to the kilometers traveled.
2. b) What is the domain of this function taking into account the context of the problem?
3. c) How much will a trip of $15$ km cost?
4. d) If a customer paid $420$ pesos, how many kilometers did they travel?

The accompanying graph represents the height in meters 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?

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?

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?

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

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

A farmer has $100$ meters 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 domain of definition of the function $f(x) = \sqrt{\frac{x - 3}{x + 2}}$. Express the result as a union of intervals.

Given the quadratic function $f(x) = x^2 + kx$, determine the value of the parameter $k$ given that the average rate of change of the function over the interval $[1, 3]$ is equal to $8$.

Analyse the following graph of the function $f(x)$ and determine:

1. a) Its domain and range.
2. b) The equations of its asymptotes.
3. c) The intervals of increase and decrease.
4. d) The coordinates of its local extrema (maximums and minimums).

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

1. a) Prove analytically whether it is an even function, an odd function, or has no symmetry.
2. b) Calculate the points of intersection with the coordinate axes.
3. c) Find the equations of all its asymptotes (vertical and horizontal).

A public parking lot has the following charges: it charges $40$ pesos for the first hour (or fraction of an hour), and $30$ pesos for each additional hour (or fraction). The maximum cost per day is $240$ pesos. Write the analytical expression of the piecewise-defined function relating the price to be paid to the parking time $t$ (in hours) for the first $5$ hours, clearly indicating the intervals of definition for each section.

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.