Rational function
A company spreads its fixed manufacturing costs across every piece it produces. The more units it makes, the smaller the fixed cost carried by each one, but this average cost never falls below a certain limit, however large the production run. A value that the function gets closer and closer to without ever reaching it, and points where the function simply "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 intercept we calculate $f(0)$, if $0$ belongs to the domain. For the $x$-axis intercepts we solve $P(x)=0$, checking that those solutions do not also make the denominator zero.
Symmetry: even and odd functions
Before sketching, it is worth checking whether the graph has any 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).
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 $f(x) = \frac{4x-1}{x+3}$.
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.
Horizontal and oblique asymptotes
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 hyperbola of inverse proportion
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.
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 $y = x$?
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.
Rational functions in real-life contexts
Many technical and economic quantities are shared out over a variable amount: a fixed cost over the units produced, a mass of solute over a growing volume, a dose over the time elapsed. All of these are rational functions, and their horizontal asymptote has a very specific meaning: the limiting value that the quantity tends to, however far the process goes.
The average production cost (in euros) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{2500 + 15x}{x}$. Calculate the average cost of…
The average production cost (in euros) of $x$ units of an electronic component is given by the function $C_m(x) = \frac{2500 + 15x}{x}$. Calculate the average cost of manufacturing 100 units and determine the value towards which the average cost tends as production becomes infinitely large, interpreting the result in the context of the problem.
A tank initially contains 500 litres 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 litres 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?
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).