Exercises: Basic concepts

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

A bicycle rental service charges a fixed fee of $\$3$ for registering for the service, plus $\$1,50$ for each 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 (ARC) of the function $f(x) = x^2 - 4$ in the intervals $[-1, 2]$ and $[2, 5]$.

Determine the domain 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 relative maxima and minima.

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 $h(x)$, indicate its domain and its range in interval notation.

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

The distance $d$ (in feet) traveled by an object in free fall 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 second ($t \in [1, 3]$), and between the third and fifth second ($t \in [3, 5]$).

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

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

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

A taxi company charges a $\$3$ base fee and $\$1,20$ for each mile traveled.

1. a) Write the function that relates the total price of the ride to the miles traveled.
2. b) What is the domain of this function taking into account the context of the problem?
3. c) How much will a ride of $15$ miles cost?
4. d) If a customer has paid $\$21$, how many miles have they traveled?

The attached graph represents the height in feet of a Ferris wheel cabin as a function of time in minutes. Observe the graph and answer:

1. a) Is it a periodic function? If so, indicate its period.
2. b) What is the maximum and minimum height reached by the cabin?
3. c) At what height is the cabin exactly $10$ minutes after it starts turning?

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

The temperature $T$ (in degrees Fahrenheit) 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 time elapsed in hours. At what instant is the maximum temperature reached? What is the value of that maximum temperature?

Find the values of the parameters $a$ and $b$ so that the following piecewise 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 $100$ feet of fencing to enclose a rectangular garden plot, making use of the straight wall of an already built warehouse, 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 garden plot as a function of the length $x$ of one of the sides perpendicular to the wall.
2. b) Determine the valid domain for $x$ in the context of this problem.
3. c) What dimensions must the garden plot have for its area to be maximum?

Determine algebraically the domain 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 $k$ knowing that the Average Rate of Change of the function in the interval $[1, 3]$ is equal to $8$.

Analyze 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 relative extrema (maxima and minima).

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

1. a) Prove algebraically whether it is an even function, an odd function or if it has no symmetry.
2. b) Calculate the x- and y-intercepts.
3. c) Find the equations of all its asymptotes (vertical and horizontal).

A public parking lot has the following rate: it charges $\$2$ for the first hour (or fraction of an hour), and $\$1,50$ for each additional hour (or fraction). The maximum cost per day is $\$12$. Write the algebraic expression of the piecewise function that relates the cost to the parking time $t$ (in hours) for the first $5$ hours, clearly indicating the domain intervals 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 that value of $a$ already calculated, also find the equations of the vertical asymptotes of the function.