Exercises: Analysing a function

Based on a function whose graph consists of two continuous sections:

• The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and has a relative maximum at $(-1, 5)$.

• The second section goes from the point $(1, -1)$ (included) to the point $(8, 3)$ (included), and has a relative minimum at $(4, -2)$.

Determine its domain and its range.

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

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

Determine analytically whether the function $f(x) = x^4 - 5x^2 + 3$ has even symmetry, odd symmetry or no symmetry.

From the following graph, state the intervals of monotonicity (increasing and decreasing) and the coordinates of its local extrema.

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

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

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

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

The graph showing the evolution of the number of weekly viewers (in thousands) of a premiere film over time (in weeks) starts at the point $(0, 10)$, increases up to $(5, 80)$, remains constant between week $5$ and week $7$, decreases down to $(10, 30)$, and remains constant between week $10$ and week $12$, where it ends at $(12, 30)$. Study the domain and the range in the context of the problem, and state the intervals in which the audience increases, decreases, or remains constant.

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

Study the symmetry of the function $f(x) = \frac{x}{x^2 + 1}$ and calculate its axis intercepts.

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

Express the modulus function $f(x) = |2x - 4|$ as a piecewise function and determine its increasing and decreasing intervals.

The variation in the water level of a reservoir $v$ (in meters relative to its average level) as a function of the month $x$ of the year is given by the function $v(x) = -0.5x^2 + 10x - 18$. Determine in which month the reservoir level is highest, what this maximum difference from the average level is, and in which months the reservoir is at exactly the average level ($v = 0$).

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 analysing its domain, its axis intercepts, its symmetry, and its horizontal and vertical asymptotes.

Given the following piecewise function, study its domain, its continuity, its axis intercepts and its intervals of monotonicity:

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

In an isosceles triangle with base $12$ cm and height $10$ cm, a rectangle is inscribed whose base $x$ lies 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 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 bloodstream, in mg/L, $t$ hours after administration is given by the model $C(t) = \frac{10t}{t^2 + 1}$. Calculate the average rate of change in the first hour (interval $[0, 1]$) and interpret the result. In addition, reason mathematically what the long-term trend of the drug concentration is (as $t \to \infty$).

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

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

• It is a function with odd symmetry.

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

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

• It passes through the origin.

You can do it on paper and send a photo with your answer.

From the quadratic function $f(x) = x^2 - 4x$, obtain the analytical expression as a piecewise function for $g(x) = |x^2 - 4x|$. Picture the situation and determine the intervals of concavity and convexity of $g(x)$.

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 meters and $t$ is the time in seconds from launch. Determine the domain of the function restricted to the actual flight duration (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 $f(x) = \frac{x^3}{x^2 - 4}$ has even symmetry, odd symmetry or neither. Then, calculate its axis intercepts and determine the equations of its vertical and horizontal asymptotes.