Rational function
A block factory in Santiago spreads its fixed costs over every piece it produces. The more blocks it makes, the less fixed cost each one carries, but the average cost never drops below a certain value, no matter how large the production. A value the function gets closer to without ever reaching it, and points where the function "breaks": that is the territory of rational functions.
Domain of a rational function
Calculate the domain of definition of the rational function $f(x) = \frac{5x-2}{x^2-9}$.
Calculate the domain of definition of the rational function $f(x) = \frac{5x-2}{x^2-9}$.
Determine the domain of the function $f(x) = \frac{x^2+1}{x^2+4x+4}$.
Determine the domain of the function $f(x) = \frac{x^2+1}{x^2+4x+4}$.
Analyse in detail the rational function $f(x) = \frac{x^2-5x+6}{x^2-4}$. Determine its domain, simplify the algebraic expression stating for which values the…
Analyse in detail the rational function $f(x) = \frac{x^2-5x+6}{x^2-4}$. Determine its domain, simplify the algebraic expression stating for which values the simplification is valid, find the equations of its asymptotes, and explain what graphical feature occurs at exactly the point with x-coordinate $x = 2$.
Intercepts and sign
For the $y$-axis, compute $f(0)$, if $0$ belongs to the domain. For the $x$-axis, solve $P(x)=0$, checking that those solutions do not also make the denominator zero.
Symmetry: even and odd functions
Before graphing, it helps to check for symmetry: if $f(-x)=f(x)$ for every $x$ in the domain, $f$ is even (symmetric about the $y$-axis); if $f(-x)=-f(x)$, it is odd (symmetric about the origin). If neither holds, there is no symmetry.
Find the coordinates of the axis intercepts of the function $f(x) = \frac{3x+12}{2x-5}$.
Find the coordinates of the axis intercepts of the function $f(x) = \frac{3x+12}{2x-5}$.
Study the sign of the rational function $f(x) = \frac{x-4}{x+2}$, clearly stating the intervals of the real number line where the function takes positive values and…
Study the sign of the rational function $f(x) = \frac{x-4}{x+2}$, clearly stating the intervals of the real number line where the function takes positive values and where it takes negative values.
Check analytically whether the function $f(x) = \frac{3x}{x^2+1}$ is even, odd or exhibits no symmetry with respect to the coordinate axes.
Check analytically whether the function $f(x) = \frac{3x}{x^2+1}$ is even, odd or exhibits no symmetry with respect to the coordinate axes.
Vertical asymptotes
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{4x-1}{x+3}$.
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{4x-1}{x+3}$.
Find the domain, the axis intercepts, and the equations of all asymptotes of the rational function $f(x) = \frac{3x^2}{x^2-16}$.
Find the domain, the axis intercepts, and the equations of all asymptotes of the rational function $f(x) = \frac{3x^2}{x^2-16}$.
Horizontal and oblique asymptotes
Above or below the oblique asymptote?
Since $f(x)=mx+n+\dfrac{r(x)}{Q(x)}$, the difference $f(x)-(mx+n)$ is exactly the remainder term. Its sign tells you where the curve is: if it is positive, the curve lies above the asymptote; if it is negative, below.
Find the equation of the oblique asymptote of the rational function $f(x) = \frac{2x^2-5x+4}{x-1}$.
Find the equation of the oblique asymptote of the rational function $f(x) = \frac{2x^2-5x+4}{x-1}$.
Determine the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{ax+7}{3x+b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at…
Determine the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{ax+7}{3x+b}$ has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = \frac{5}{3}$.
Given the function $f(x) = \frac{x^2+1}{x}$, determine its domain, its asymptotes (vertical, horizontal, or oblique), and study the position of the curve with respect to…
Given the function $f(x) = \frac{x^2+1}{x}$, determine its domain, its asymptotes (vertical, horizontal, or oblique), and study the position of the curve with respect to its oblique asymptote as $x \to +\infty$ and as $x \to -\infty$.
The inverse proportionality hyperbola
Given the function $f(x) = \frac{1}{x+2} - 4$, explain what elementary geometric transformations (horizontal and vertical translations) have been applied to the graph of…
Given the function $f(x) = \frac{1}{x+2} - 4$, explain what elementary geometric transformations (horizontal and vertical translations) have been applied to the graph of the basic hyperbola $y = \frac{1}{x}$ to obtain it.
Determine the values of the real parameter $m$ for which the line with equation $y = mx$ intersects the rational curve $y = \frac{1}{x-2}$ at exactly a single point.…
Determine the values of the real parameter $m$ for which the line with equation $y = mx$ intersects the rational curve $y = \frac{1}{x-2}$ at exactly a single point. Geometrically interpret what these specific lines represent for the curve.
From the following graph of a rational function, determine its domain, the equations of its asymptotes (vertical and horizontal), and the axis intercepts. Next, suggest…
From the following graph of a rational function, determine its domain, the equations of its asymptotes (vertical and horizontal), and the axis intercepts. Next, suggest an algebraic expression of the type $f(x) = \frac{ax^2}{x^2+b}$ that perfectly matches all these visual features.
Equations, inverse and composition
To find out when $f(x)$ equals a given number, you set them equal and solve the equation. Multiply by the denominator, solve, and check that the solution is in the domain.
Inverse function
The inverse function is obtained by swapping $x$ and $y$ and solving for $y$. Its domain is the range of $f$.
Composition: computing $f(f(x))$
The composition $(f\circ f)(x)=f(f(x))$ is computed by substituting the expression for $f$ in place of the $x$ in $f$. Then you simplify the complex fraction by multiplying the numerator and denominator by the inner denominator.
If $(f\circ f)(x)=x$, then $f$ is its own inverse: its graph is symmetric about the line $y=x$ (the bisector). Apply the same method to $f(x)=\dfrac{x+2}{x-1}$ in the exercise below.
Find the value of $x$ for which the function $f(x) = \frac{2x+3}{x-1}$ takes exactly the value $f(x) = 5$.
Find the value of $x$ for which the function $f(x) = \frac{2x+3}{x-1}$ takes exactly the value $f(x) = 5$.
Calculate the algebraic expression for the inverse function of $f(x) = \frac{2x-1}{x+3}$ and specify the domain of this new function $f^{-1}(x)$.
Calculate the algebraic expression for the inverse function of $f(x) = \frac{2x-1}{x+3}$ and specify the domain of this new function $f^{-1}(x)$.
Let $f(x) = \frac{x+2}{x-1}$ be a rational function. Prove analytically that the composite function $(f \circ f)(x) = x$ for every value of $x$ belonging to its domain.…
Let $f(x) = \frac{x+2}{x-1}$ be a rational function. Prove analytically that the composite function $(f \circ f)(x) = x$ for every value of $x$ belonging to its domain. What geometric conclusion about the graph of $f(x)$ can you draw from this result in relation to the line $y = x$?
Parameters in the function
When the function contains letters, the conditions in the problem (asymptotes, points on the graph) turn into equations. For example, a vertical asymptote at $x=2$ for $\dfrac{ax+7}{3x+b}$ requires $3(2)+b=0$; the horizontal one comes from the quotient of leading coefficients, $a/3$. For an oblique asymptote, divide and compare the quotient with the given line.
Parameter in the denominator
If the parameter $k$ is in a quadratic denominator, the vertical asymptotes depend on its real roots, and the discriminant $\Delta=b^2-4ac$ tells you how many there are. First, check that those roots do not also make the numerator zero.
Discuss, depending on the real parameter $k$, the existence of vertical asymptotes for the function $f(x) = \frac{x^2-2x+5}{x^2-kx+9}$. Determine the interval of values…
Discuss, depending on the real parameter $k$, the existence of vertical asymptotes for the function $f(x) = \frac{x^2-2x+5}{x^2-kx+9}$. Determine the interval of values of $k$ for which the function has no vertical asymptotes at all.
Find the values of the parameters $a$, $b$ and $c$ of the rational function $f(x) = \frac{ax^2+bx+c}{x-3}$ given that it has an oblique asymptote with equation…
Find the values of the parameters $a$, $b$ and $c$ of the rational function $f(x) = \frac{ax^2+bx+c}{x-3}$ given that it has an oblique asymptote with equation $y = 2x+1$ and that its graph passes through the point with coordinates $(4, 15)$.
Bounds and growth without derivatives
Sometimes you need to prove that a rational function does not exceed a certain value, or that it always increases. This can be done with algebra alone.
Bounding with the discriminant
Set $f(x)=C$, turn it into a quadratic equation in $x$, and require $\Delta\ge0$ so that it has a real solution. That limits the possible values of $C$.
Bounding with a square
Another way: $(t-a)^2\ge0$ gives $t^2+a^2\ge2at$. Dividing by $t^2+a^2>0$ gives $\dfrac{2at}{t^2+a^2}\le1$, with equality at $t=a$.
Growth: divide and compare
If you divide, $f=A-\dfrac{B}{t^2+1}$ and you can reason without derivatives.
The concentration $C(t)$ of a drug in the bloodstream $t$ hours after intramuscular administration is given by the expression $C(t) = \frac{30t}{t^2+9}$ (measured in…
The concentration $C(t)$ of a drug in the bloodstream $t$ hours after intramuscular administration is given by the expression $C(t) = \frac{30t}{t^2+9}$ (measured in mg/L). Without using differentiation tools, prove algebraically that the maximum concentration can never exceed $5$ mg/L and determine the exact time $t$ at which this maximum concentration is reached. (Hint: Set up the equality $C = \frac{30t}{t^2+9}$ as a quadratic equation in the variable $t$ and analyse the condition that its discriminant must satisfy for the time to be a real number).
The population of a species of fish in a protected lake follows a growth model given by the function $P(t) = \frac{2000t^2+500}{t^2+1}$, where $t \ge 0$ is measured in…
The population of a species of fish in a protected lake follows a growth model given by the function $P(t) = \frac{2000t^2+500}{t^2+1}$, where $t \ge 0$ is measured in years from the start of the study. Determine the initial population, the population in the long term (limit as $t \to +\infty$) and prove algebraically that the population is always growing over time. (Hint: Try to express the fraction in the form $A - \frac{B}{t^2+1}$ by carrying out polynomial division).
Rational functions in real-life contexts
Many quantities are shared out over a varying amount: a fixed cost over the units produced, salt over a growing volume, a dose over time. All of them are rational functions, and their horizontal asymptote is the limiting value the quantity approaches in the long run.
Other situations require solving an equation (for example, the moment when a concentration reaches a certain value): you set up the equality, clear the denominator, and check that the solution is valid in the context.
Back to the beginning: the average cost at the block factory approaches its variable cost per unit (the horizontal asymptote) without ever reaching it.
The average production cost (in pesos) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{150\,000 + 900x}{x}$. Calculate the average cost…
The average production cost (in pesos) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{150\,000 + 900x}{x}$. Calculate the average cost of manufacturing 100 units and determine the value toward which the average cost tends as production becomes infinitely large, interpreting the result in the context of the problem.
A tank initially contains 500 liters of pure water. A saline solution with a concentration of 40 g/L is pumped in at a rate of 10 L/min. The salt concentration in the…
A tank initially contains 500 liters of pure water. A saline solution with a concentration of 40 g/L is pumped in at a rate of 10 L/min. The salt concentration in the tank after $t$ minutes is given by the function $C(t) = \frac{400t}{500+10t}$. At what time will the concentration be exactly 25 g/L? What is the equation of the horizontal asymptote of this function and what physical meaning does it have?
Two workers work together to paint a wall. The younger worker would take $x$ hours to do it working alone, while the older worker would take $x+3$ hours. The total time…
Two workers work together to paint a wall. The younger worker would take $x$ hours to do it working alone, while the older worker would take $x+3$ hours. The total time $T(x)$ it takes them when working simultaneously is given by the function $T(x) = \frac{x(x+3)}{2x+3}$. If working together they manage to finish the job in exactly 2 hours, how long would it have taken the younger worker to paint the wall on his own?