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 $f(x) = \frac{5}{x - 3} + 2$.

Seán has to study the axes intercepts of the rational function $f(x) = \frac{x^2 - 4}{x - 2}$. On his exam, he writes the following procedure:

«1. $Y$-axis intercept: I calculate $f(0) = \frac{0^2 - 4}{0 - 2} = \frac{-4}{-2} = 2$, so the graph intersects at the point $(0, 2)$.

1. $X$-axis intercepts: I solve $f(x) = 0 \implies x^2 - 4 = 0 \implies x = 2$ and $x = -2$. Therefore, the graph intersects the $X$-axis at two points: $(2, 0)$ and $(-2, 0)$».

1. a) Identify the conceptual error made by Seán: what fundamental step in the outline for studying a function did he omit before finding the intercepts, and why does this omission invalidate one of his conclusions?
2. b) Determine the real domain of the function $f(x)$ and write the corrected list of the intercepts that actually exist.
3. c) Based on the above, reason whether a vertical asymptote exists at the point $x = 2$ or if the function exhibits a different behavior, simplifying the formula for $x \neq 2$.

Consider the rational function $f(x) = \frac{6x - 2}{2x + 1}$.

1. a) Before carrying out any precise calculation, make an estimate: when the values of $x$ become extremely large (for example, $x$ close to one million), the constant terms ($-2$ and $+1$) barely influence the result. Taking this into account, what simple numerical value do you think the images $f(x)$ will approach? What line do you suspect will be the horizontal asymptote?
2. b) Check numerically whether your estimate was correct by calculating the exact values of $f(10)$, $f(100)$, and $f(1\,000)$ (round to three decimal places if necessary).
3. c) From your checks, deduce the equation of the horizontal asymptote both as $x \to +\infty$ and as $x \to -\infty$, and explain what visual information this asymptote provides about the shape of the curve at the edges of the paper.

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 metres 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 metres 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 behaviour 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.

Based on the records from a Met Éireann station located in the Irish Midlands during a clear autumn day, the temperature $T$ (in degrees Celsius, ºC) throughout the $24$ hours of the day is modeled by the quadratic function:

$T(t) = -0.05 \times (t - 15)^2 + 14$, where $t$ is the time elapsed in hours since midnight ($0 \leq t \leq 24$).

1. a) Justify the domain of definition of this function considering the physical context of the problem.
2. b) Without differentiating, and based on the structure of the vertex of the parabola, identify with reasoning the time interval during which the temperature rises, the interval during which it falls, and the full coordinates $(t, T)$ of the relative extremum (daily maximum).
3. c) Calculate the average rate of change (ARC) of the temperature in the morning period between $t = 7$ hours and $t = 11$ hours. Interpret the real meaning of the sign and the units of the resulting quantity.
4. d) Set up and solve the equation needed to find out whether $0$ ºC was reached at any point during the day (intercepts with the horizontal time axis).

To study whether the rational function $g(x) = \frac{x^3}{x^2 - 9}$ has any type of symmetry (even, odd, or neither), three students propose different approaches:

• Aoife proposes testing it using numerical values: calculating pairs of opposite images such as $g(1)$ and $g(-1)$, $g(2)$ and $g(-2)$, $g(4)$ and $g(-4)$, and checking whether they match or change sign.

• Seán proposes applying the formal algebraic definition: replacing $x$ with $-x$, calculating the expression of $g(-x)$, simplifying it, and directly comparing it with $g(x)$ and with $-g(x)$.

• Niamh proposes decomposing the function: analyzing separately that the numerator is an odd function and the denominator is an even function, and deducing the overall symmetry by applying a sign rule analogous to the multiplication and division of numbers.

1. a) Critically analyze the three methods: which one or ones constitute a valid mathematical proof for all numbers in the domain, and which one only serves as an indication or experimental check?
2. b) Apply Seán's algebraic method to rigorously prove what symmetry $g(x)$ possesses.
3. c) Explain in detail how the result of the symmetry allows you to save half the work when studying the vertical asymptotes and the sign of the function on its domain.

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 upwards and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in metres 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.

Find the analytic expression of a rational function of the form $f(x) = \frac{a \times x + b}{x - c}$, where $a$, $b$, and $c$ are real numbers that you must determine, knowing that it simultaneously satisfies the following three conditions:

1. Its graph has a vertical asymptote at the line $x = 3$.

2. It intersects the $X$-axis at the point $(-2, 0)$.

3. When the values of the independent variable grow without bound ($x \to +\infty$), the images approach the horizontal line with equation $y = 4$.

1. a) Deduce step by step with justification the value of the coefficients $a$, $b$, and $c$. Check that the numerator of the resulting function does not vanish at $x = 3$.
2. b) Find the $Y$-axis intercept of the graph.
3. c) Calculate the average rate of change (ARC) of the function on the intervals $[4, 5]$ and $[5, 8]$. In light of these results and the vertical asymptote found, reason whether it is possible for the function to have any relative maximum or minimum on the interval $(3, +\infty)$ or if its monotonicity is constant on it.

An agricultural cooperative in County Wexford, Ireland, acquires a new packaging machine through a commercial lease. The machine generates monthly fixed costs of $1\,200$ € (insurance and installment fee) and a variable cost of $25$ € per ton of fruit processed. The function expressing the average unit cost per ton (in euros per ton) when processing $x$ tons in a month is:

$\bar{C}(x) = \frac{1\,200 + 25x}{x}$

1. a) Determine the domain of definition of $\bar{C}(x)$ in the real context of the problem and explain why values $x \leq 0$ lack operational meaning.
2. b) Analyze the trend and the vertical asymptote of the function as production becomes very close to zero from the right ($x \to 0^+$). What economic interpretation does it have that the value of $\bar{C}(x)$ grows toward $+\infty$ when a minuscule amount of fruit is processed?
3. c) Calculate analytically the horizontal asymptote of the function as $x \to +\infty$. Reason whether the cooperative will ever be able to reduce its average cost below $25$ € per ton by increasing the production volume indefinitely.
4. d) Prove that the function is strictly decreasing throughout its domain by showing that $ARC[a, b] = -\frac{1\,200}{ab} < 0$ for any $0 < a < b$; then evaluate the ARC on two intervals of equal length but different production scale (for example, $[10, 30]$ and $[100, 120]$), and interpret what practical conclusion the cooperative's management should draw regarding the amortization of fixed costs.