Exercises: Analysing a function

Look at the following graph and determine its domain and range.

Calculate the average rate of change of the function $f(x) = 2x^2 - 3x$ on the interval $[-1, 2]$.

Find the x- and y-intercepts of the function $f(x) = \frac{3x - 6}{x + 2}$.

Determine analytically whether the function $f(x) = x^4 - 5x^2 + 3$ is even, odd, or neither.

From the following graph, determine the intervals where the function is increasing and decreasing, and find the coordinates of its relative extrema.

Find the domain of the radical function $f(x) = \sqrt{2x - 8}$.

Analyze the continuity of the following piecewise function at the point $x = 1$:

$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$

Find the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{5}{x - 3} + 2$.

The following graph shows the growth of a bacterial population (in thousands) cultured in a laboratory over time (in hours). Analyze the domain and range in the context of the problem, and indicate on which intervals the population is increasing, decreasing, or constant.

Find the domain of the rational function $f(x) = \frac{x - 1}{x^2 - 4x - 5}$.

Analyze the symmetry of the function $f(x) = \frac{x}{x^2 + 1}$ and find its x- and y-intercepts.

The position of an object in linear motion is given by the function $e(t) = 5t^2 + 2t$, where $e$ is the distance in feet and $t$ is the time in seconds. Calculate the average speed (average rate of change) on the intervals $[1, 3]$ and $[3, 5]$, and indicate in which interval the object moved faster.

Express the absolute value function $f(x) = |2x - 4|$ as a piecewise function and determine the intervals on which it is increasing and decreasing.

The profit of a startup company, in thousands of dollars, is given by the function $B(x) = -0.5x^2 + 10x - 18$, where $x$ represents the years elapsed since its founding. Determine in which year the maximum profit is reached, how much this profit is, and in which years the company breaks even (zero profit).

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

Describe the behavior of the function $f(x) = \frac{2x^2}{x^2 + 1}$ by analyzing its domain, its x- and y-intercepts, its symmetry, and its horizontal and vertical asymptotes.

Given the following piecewise function, analyze its domain, its continuity, its x- and y-intercepts, and the intervals where it is increasing and decreasing:

$f(x) = \begin{cases} x^2 - 4 & \text{if } x < 0 \\ 2x - 4 & \text{if } 0 \leq x < 4 \\ 4 & \text{if } x \geq 4 \end{cases}$

In an isosceles triangle with a base of $12$ in and a height of $10$ in, a rectangle is inscribed whose base $x$ rests on the base of the triangle, as shown in the figure. Express the area of the rectangle as a function of its base $x$ and determine the domain of this function in the geometric context of the problem.

The rational function $f(x) = \frac{ax + b}{x + c}$ passes through the point $(0, 1)$, has a vertical asymptote at the line $x = 2$, and has a horizontal asymptote at the line $y = 3$. Set up a system of equations and find the values of the parameters $a$, $b$, and $c$.

The concentration of a certain drug in a patient's blood, in mg/L, $t$ hours after its administration is given by the model $C(t) = \frac{10t}{t^2 + 1}$. Calculate the average rate of change over the first hour (interval $[0, 1]$) and interpret the result. Furthermore, explain mathematically what the long-term trend of the drug's concentration is (as $t \to \infty$).

Sketch the graph of a function $f(x)$ that simultaneously satisfies all the following conditions:

• Its domain is $\mathbb{R} \setminus \{-2, 2\}$.

• It is an odd function.

• It has vertical asymptotes at $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 0$.

• It has a relative maximum at the point $(-4, 1)$ and a relative minimum at $(4, -1)$.

• It passes through the origin.

From the quadratic function $f(x) = x^2 - 4x$, obtain the analytical expression as a piecewise function of $g(x) = |x^2 - 4x|$. Visualize the situation and determine the intervals where $g(x)$ is concave up and concave down.

A projectile is launched upward and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in feet and $t$ is the time in seconds since launch. Determine the domain of the function restricted to the actual duration of the flight (until it hits the ground), the height from which it was launched, the maximum height it reaches, and the total flight time.

Prove analytically whether the function $f(x) = \frac{x^3}{x^2 - 4}$ is even, odd, or neither. Then, find its x- and y-intercepts and determine the equations of its vertical and horizontal asymptotes.