Exercises: Types of functions

Given the polynomial function $f(x) = x^3 - 3x^2 + 4$, carry out a basic study of its behaviour following these steps:

1. a) Calculate the axis intercepts.
2. b) Find the intervals of increase and decrease using the first derivative $f'(x)$.
3. c) Determine the coordinates $(x, y)$ of its relative maxima and minima.

Consider the rational function $f(x) = \frac{2x + 4}{x - 3}$. To understand its structure before graphing it, analyze its main elements. The figure shows part of its graph; the tick marks on the axes are 1 unit apart.

1. a) Determine its domain.

1. b) Calculate the equations of its vertical and horizontal asymptotes.

1. c) Study the symmetry of the function. Is it even, odd, or does it show no symmetry?

Functions with absolute values often have points where the graph changes direction abruptly. Let $f(x) = |x - 4|$ be a function.

1. a) Express the function as a piecewise-defined function.

1. b) State the point on the graph where the function is continuous but not differentiable (corner point).

1. c) Sketch its graph, indicating the vertex.

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

The domain of a function is fundamental for sketching its graph. Find the domain of the following transcendental functions:

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

Briefly explain why restrictions exist in each case.

A biologist studies the evolution of a bacteria population in a controlled environment. The number of bacteria (in thousands) is given by the function $P(t) = \frac{40t}{t + 2}$, where $t$ is the time in hours since the start of the experiment ($t \geq 0$). The figure shows the graph of $P(t)$.

1. a) What is the population at the start of the experiment ($t = 0$)?

1. b) What is the limit of the population as time grows indefinitely? Interpret the result in terms of asymptotes.

1. c) Does the population increase or decrease over time? Justify your answer using the derivative $P'(t)$.

Given the function $f(x) = x^4 - 2x^2$, study its curvature by following these steps:

1. a) Calculate the second derivative $f''(x)$.
2. b) Find the intervals of concavity ($\cup$) and convexity ($\cap$).
3. c) Find the points of inflection of the function.

Determine the parity of the following functions, justifying mathematically whether they are even ($f(x) = f(-x)$), odd ($f(-x) = -f(x)$), or neither:

1. a) $f(x) = x^2 \times \cos(x)$
2. b) $g(x) = \frac{x^3}{x^2 + 1}$
3. c) $h(x) = e^x - x$

Look at the following graph of a function $f(x)$ (the dashed lines are the asymptotes of the function; use the tick marks on the axes, one per unit) and extract the requested information:

1. a) State the values of $x$ where the vertical asymptotes are located.

1. b) Estimate the intervals where the function is increasing.

1. c) Identify the value of $\lim_{x \to +\infty} f(x)$ by observing the behavior of the graph.

Cian has to calculate the relative extrema of the absolute value function $f(x) = |x^2 - 9|$. On his exam, he writes the following:

"Since the absolute value does not change the degree of the polynomial, I directly differentiate its argument to obtain the slope: $f'(x) = 2x$. Setting it equal to zero, I get $2x = 0 \implies x = 0$. Substituting, $f(0) = |-9| = 9$. Since $f''(0) = 2 > 0$, I conclude that the function has a single relative extremum: a minimum at the point $(0, 9)$".

1. a) Identify the two serious conceptual errors made by Cian by ignoring the definition of absolute value and the differentiability of the function.
2. b) Write $f(x)$ correctly as a piecewise-defined function according to the sign of its argument.
3. c) Find all the real relative extrema of $f(x)$, reasonably classifying them as maxima or minima and distinguishing which ones correspond to smooth points ($f'(x) = 0$) and which ones to corner points or cusps where the function is not differentiable.

To study the asymptotes and the relative position of the curve of the rational function $f(x) = \dfrac{2x^2 + 3x - 1}{x + 2}$, two students propose different strategies:

• Method A (Alba): Perform polynomial division between the numerator and denominator to decompose the function into the form $f(x) = c(x) + \dfrac{r(x)}{d(x)}$.

• Method B (Bruno): Apply the classical limits $m = \lim_{x \to \pm\infty} \dfrac{f(x)}{x}$ and $n = \lim_{x \to \pm\infty} [f(x) - m \times x]$.

1. a) Compare both methods and justify why Alba's method is more efficient in this case to simultaneously determine the equation of the oblique asymptote and the position of the graph relative to it at $+\infty$ and $-\infty$ without needing to calculate additional limits.
2. b) Apply Alba's method to find the asymptote line and analytically determine whether the curve lies above or below this asymptote at each of the two infinities.

Given the rational function $f(x) = \frac{2x^2}{x^2 - 9}$, carry out a complete study for its graphical representation following these steps:

1. a) Find the domain of definition and the points where the graph crosses the coordinate axes.

1. b) Study whether the function has any symmetries.

1. c) Determine the vertical and horizontal asymptotes and the position of the curve with respect to them.

1. d) Calculate the intervals of increase and decrease, and determine the relative maxima and minima.

1. e) Sketch the graph of the function based on the previous results.

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

Consider the function $f(x) = \frac{x^2 + 3}{x - 1}$. Find:

1. Find the equations of its asymptotes. If an oblique asymptote exists, determine the position of the curve with respect to it as $x \to +\infty$ and $x \to -\infty$.

2. Find the intervals of concavity and convexity, and check whether there are points of inflection.

3. Use the information on increasing/decreasing intervals and extrema to draw an accurate sketch of the graph.

A biological study on an insect population in a controlled environment follows the function $P(t) = \frac{50t}{t^2 + 4} + 2$, where $P$ is the population in thousands and $t$ is the time in months from the start of the experiment ($t \geq 0$).

1. Determine at what time the population reaches its maximum value and what that value is.

2. What value does the population approach if the experiment were to continue indefinitely? Justify your answer by evaluating limits.

3. Study the curvature of the function to identify when the population decline begins to slow down (point of inflection).

Given the absolute value function $f(x) = |x^2 - 4x + 3|$:

1. a) Express the function as a piecewise-defined function.

1. b) Study the differentiability of the function at the points $x = 1$ and $x = 3$. What happens geometrically at those points?

1. c) Find the relative extrema of the function and graph it.

Let $f(x) = (x - 2)e^x$ be the function. Carry out a detailed study of it in order to sketch its graph, including:

1. Domain and axis intercepts.

2. Horizontal asymptotes and behaviour at infinity.

3. Intervals of monotonicity and relative extrema.

4. Points of inflection and curvature.

Analyse and sketch the logarithmic function $f(x) = \ln(x^2 + 1)$. In your analysis, you must include:

1. Proof that the function is even. How does this affect its graphical representation?

2. Determination of the intervals of increase and decrease.

3. Study of the curvature and finding the points of inflection.

4. Does the function have any type of asymptotes? Justify your answer.

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

Consider the piecewise function:

$$f(x) = \begin{cases} \frac{1}{x-2} & \text{if } x < 0 \\ x^2 - 2x - \frac{1}{2} & \text{if } x \geq 0 \end{cases}$$

1. Study the continuity and differentiability of the function at $x = 0$.

2. Find the asymptotes of the function in its domain of definition.

3. Calculate the local minima and maxima of each section in its corresponding interval.

4. Sketch the graph of the function.

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

Determine the values of the parameters $a$ and $b$ so that the function $f(x) = ax^3 + bx^2 - x + 4$ has an inflection point at $x = 1$ and the slope of the tangent line at that point is $1$. Once you have found the parameters:

1. a) Study the monotonicity of the resulting function.

1. b) Determine whether it has any relative maximum or minimum.

1. c) Justify whether the function has asymptotes.

Given the function $f(x) = \dfrac{\ln(x)}{x}$, answer the following questions by combining logical deduction and formal calculation:

1. Estimation phase (without differentiating or calculating formal limits yet):

a) From the domain of the logarithmic function and knowing that $\ln(x) \to -\infty$ as $x \to 0^+$, estimate the behavior of the curve near the $Y$-axis. Will there be a vertical asymptote? Where will the branch head towards?

b) The notes emphasize that the logarithm "grows much slower than any polynomial". Based on this, estimate what value $f(x)$ will approach as $x \to +\infty$ and deduce whether a horizontal asymptote will exist.

c) Knowing that $f(1) = 0$, that $f(x) < 0$ for $x \in (0, 1)$, and that for $x > 1$ the function is positive but must approach the horizontal asymptote as $x \to +\infty$, deduce whether at least one relative maximum must necessarily exist and estimate in which interval of $x$ it should be located.

1. Verification phase:

d) Analytically confirm your estimates by calculating $\lim_{x \to 0^+} f(x)$ and $\lim_{x \to +\infty} f(x)$.

e) Calculate the derivative $f'(x)$, determine the exact coordinates $(x, y)$ of its relative extremum, and check whether your previous deduction regarding the existence and position of the maximum was correct.

During a winter thermal inversion episode in Dublin, the concentration of nitrogen dioxide ($NO_2$) in the air, measured in micrograms per cubic meter ($\mu\text{g/m}^3$), is monitored over a full day ($t \in [0, 24]$ hours). The air quality department models its evolution using a piecewise function that links the morning accumulation phase due to traffic with the evening dispersion:

$$C(t) = \begin{cases} 40 + 10t - 0.5t^2 & \text{if } 0 \le t \le 12 \\ 124 \times e^{-0.1(t-12)} - 36 & \text{if } 12 < t \le 24 \end{cases}$$

1. a) Check whether the model guarantees continuity at the transition point ($t = 12$). Next, study differentiability at that point and determine whether the graph exhibits a corner point or a smooth transition. Interpret the physical meaning of this fact.
2. b) Find at what time of day the maximum pollutant concentration is reached and calculate its exact value.
3. c) If the exponential dispersion model were to continue indefinitely ($t \to +\infty$), would it have a horizontal asymptote? Analytically justify why the exponential term collapses to zero and what the theoretical residual pollution level would be according to that asymptote.

Consider the piecewise-defined function:

$$f(x) = \begin{cases} \dfrac{x^2 - 4}{x+2} & \text{if } x < -2 \\ a \times e^{x+2} + b x & \text{if } -2 \le x < 0 \\ \dfrac{x+1}{x^2+1} & \text{if } x \ge 0 \end{cases}$$

Carry out a complete study of the function following these steps:

1. Find the values of $a$ and $b$ so that the function is continuous at $x = -2$ and has a relative extremum at $x = -1$.

2. For the values found, study differentiability at $x = -2$ and $x = 0$.

3. Calculate the asymptotes of the function on its entire domain.

4. Determine the intervals of increase and decrease, as well as the relative maxima and minima on the interval $(-\infty, 0)$.

Consider the rational function defined by:

$$f(x) = \frac{x^3 + ax^2 + b}{x^2 - 1}$$

Determine the values of the real parameters $a$ and $b$ knowing that the line $y = x + 2$ is an oblique asymptote of the function and that its graph passes through the point $(0, 0)$. Once the parameters are calculated:

1. Find the domain and the vertical asymptotes, studying the position of the curve relative to them.

2. Find the intercepts with the coordinate axes.

3. Study the existence of inflection points. Is it possible for this function to have no inflection points in its domain? Justify your answer.

A communications satellite describes a trajectory whose signal intensity received on Earth is given by the function:

$$S(t) = \ln |t^2 - 9| - \frac{t}{2}$$

where $t$ represents the time in hours since the start of tracking ($t \ge 0$).

1. Determine the domain of the function and identify the times at which the signal is completely interrupted.

2. Find the vertical asymptotes and explain their meaning in this context.

3. Calculate the intervals where the signal intensity is increasing and decreasing.

4. Determine at what time the signal reaches its relative maximum value and sketch the graph of the intensity trajectory.

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

Given the function:

$$f(x) = \frac{x \times |x|}{x - 1}$$

1. Express the function as a piecewise defined function by removing the modulus.

2. Study the continuity and differentiability of the function in its entire domain, paying special attention to the point $x = 0$.

3. Find the asymptotes of the graph of $f$.

4. Determine the intervals where the function is increasing and decreasing and the relative extrema.

5. Sketch the graph of the function, clearly indicating the points where the derivative does not exist, if any.

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

Consider the function $f(x) = (x-2)^2 \times e^x$.

1. Calculate the axis intercepts and the asymptotes of the function.

2. Determine the intervals of monotonicity and local extrema (maximums and minimums).

3. Find the intervals of curvature and the points of inflection.

4. Knowing that $e^{-2} \approx 0.135$ and $e^0 = 1$, sketch the graph of the function.

5. Calculate the area of the triangle formed by the origin of coordinates and the points on the graph where the tangent is horizontal.

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

Study and graph the "mixed-type" function:

$$f(x) = x \times e^{1/x}$$

To do so, analyze in detail:

1. a) The domain of definition and the behavior of the function near $x = 0$, calculating the one-sided limits $\lim_{x \to 0^+} f(x)$ and $\lim_{x \to 0^-} f(x)$.

1. b) The vertical and oblique asymptotes.

1. c) The existence of relative extrema using the first derivative.

1. d) The concavity and convexity by studying the sign of $f''(x)$.

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

We wish to design the profile of a ramp for a skatepark. The side profile follows the curve of the function:

$$f(x) = \frac{\ln(x^2+1)}{x}$$

for $x \in [-5, 5], x \neq 0$. Find:

1. Show that the function is odd. What does this imply for its graph?

2. Define the value of $f(0)$ so that the function is continuous at $x=0$. To do this, calculate $\lim_{x \to 0} f(x)$ (you may use L'Hôpital's rule).

3. Is the function differentiable at $x=0$ with the previous definition? Justify your answer.

4. Determine the points on the ramp where the slope is zero.

5. Investigate whether there are horizontal asymptotes when the domain is extended to the whole real line.

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

• Its domain is $\mathbb{R} - \{1\}$.

• It has a vertical asymptote at $x = 1$ such that $\lim_{x \to 1^-} f(x) = +\infty$ and $\lim_{x \to 1^+} f(x) = -\infty$.

• It has an oblique asymptote $y = x$ as $x \to +\infty$.

• It has a horizontal asymptote $y = -2$ as $x \to -\infty$.

• $f'(x) < 0$ on the interval $(-\infty, -2)$ and $f'(x) > 0$ on $(-2, 1) \cup (1, +\infty)$.

• It has a relative minimum at the point $(-2, -4)$.

• $f''(x) > 0$ on $(-3, 1)$ and $f''(x) < 0$ on $(-\infty, -3) \cup (1, +\infty)$.

• It intersects the $x$-axis only at the points $x = 0$ and $x = 2$.

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

You want to design the analytic expression of an irreducible rational function $f(x) = \dfrac{P(x)}{Q(x)}$ that simultaneously satisfies the following four geometric and analytic conditions:

1. Its domain is all of $\mathbb{R}$ (it has no vertical asymptote).

2. It is symmetric with respect to the origin (odd function).

3. The line $y = 0$ is its horizontal asymptote both at $+\infty$ and at $-\infty$.

4. It reaches a relative extremum at the point $(2, 4)$.

1. a) Justify theoretically why a polynomial function could never satisfy conditions 1, 3 and 4 at the same time. To do so, recall how a polynomial behaves at infinity: if it is not constant, its limit as $x\to\pm\infty$ is $+\infty$ or $-\infty$ (determined by the highest-degree term); and if it is constant, condition 3 forces it to be the zero polynomial.

1. b) Analyze with reasoning what minimum degrees the polynomials $P(x)$ and $Q(x)$ must have, and what parity the exponents of each must have, to guarantee that conditions 1, 2 and 3 are met.

1. c) Deduce the exact algebraic expression of the simplest function that satisfies all the given conditions.

1. d) Prove using the first derivative whether the point $(2, 4)$ is a relative maximum or minimum, and determine, by symmetry alone and without doing any new derivative calculations, the coordinates and nature of the other relative extremum of the function.

To analyze the relative extrema and the asymptotic behavior of the function $f(x) = |x^2 - 4| \times e^{-x}$, two 6th Year (Leaving Certificate) students propose different methods:

• Path A (Analytical piecewise decomposition): Study the sign of the argument $x^2 - 4$, rewrite $f(x)$ as a piecewise function removing the absolute value bars, calculate the derivative of each piece separately, and analyze the transition points using one-sided limits of the derivatives.

• Path B (Geometric transformation on an auxiliary function): Fully study the differentiable auxiliary function $g(x) = (x^2 - 4) \times e^{-x}$ (axis intercepts, extrema by setting a single derivative equal to zero, and asymptotes) and, subsequently, obtain the graph and properties of $f(x) = |g(x)|$ through a reflection across the $X$-axis of the regions where $g(x) < 0$.

1. a) Critically compare both paths. Justify why Path B is conceptually more powerful for identifying corner points (cusps) and deducing the new position and classification of the extrema without differentiating piecewise functions.
2. b) Following Path B, calculate the derivative of the auxiliary function $g(x)$ and find the exact $x$-coordinates of its relative extrema.
3. c) From the behavior of $g(x)$, determine with reasoning for the original function $f(x)$:

• The coordinates of all its relative extrema, expressly indicating which are smooth points ($f'(x) = 0$) and which are cusps where the derivative does not exist.

• The horizontal asymptotes at $+\infty$ and at $-\infty$, justifying why the function collapses towards zero in only one of the two infinite directions.

Unit challenge. How does our country's energy consumption breathe? Your mission is to model and analytically analyse the daily electricity demand curve for a real day.

1. Data collection: Go to the EirGrid Smart Grid Dashboard and collect yesterday's real all-island demand (in GW) hour by hour, for $t \in [0, 24]$.

2. Modelling and classification: Enter the ordered pairs into GeoGebra and propose a function $f(t)$ that approximates the overall behaviour (a polynomial of degree 3 or 4, or a rational function). Classify the chosen type of function and justify your decision.

3. Analytical study: Analytically extract all the information from $f(t)$: determine the domain, asymptotes (if any), intervals of monotonicity, and extrema using $f'(t)=0$, as well as curvature and inflection points with $f''(t)=0$.

4. Conclusions report: Prepare a brief report (one page) with the graph overlaid on the data, comparing your analytical maxima and minima with the actual consumption peaks (peak and off-peak hours).

Assessment criteria:

• Rigour in the analytical study of the function (derivatives, monotonicity, curvature, and asymptotes).

• Appropriate justification for the type of function and the fit in GeoGebra.

• Coherence in the interpretation within the report when comparing the model with real data.