Characteristics of Functions
Yaritza analyzes the graph of a corner store's profits over the week: when are there losses? When is the most money made? Does the pattern repeat every week? To answer, we always ask a function the same questions: Where is it defined? Where does it cross the axes? Where is it positive or negative? What values does it take? Does it go up or down? At the end of the page we will come back to the corner store's profits.
Domain with several restrictions at once
Determine the domain of the rational function $f(x) = \frac{x-1}{x^2-9}$.
Determine the domain of the rational function $f(x) = \frac{x-1}{x^2-9}$.
Find the domain of the radical function $g(x) = \sqrt{2x-8}$.
Find the domain of the radical function $g(x) = \sqrt{2x-8}$.
Find the domain of the logarithmic function $s(x) = \log_3(x+2)$.
Find the domain of the logarithmic function $s(x) = \log_3(x+2)$.
Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.
Find the domain of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.
Determine the domain of the function $f(x) = \ln(x^2 - 4x + 3)$.
Determine the domain of the function $f(x) = \ln(x^2 - 4x + 3)$.
Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, justifying your answer.
Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, justifying your answer.
Domain in a context
In a problem, besides the mathematical conditions, the context also limits $x$: a length, a number of units, or a time cannot be negative. For example, if $x$ is the number of units produced, we require $x>0$. If $A(x)=x(100-2x)$ is the area of a garden, we need $x>0$ and $100-2x>0$, that is, $0
Intercepts and sign analysis
Find the axis intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.
Find the axis intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.
Determine the intervals in which the function $q(x) = -3x + 12$ is positive and those in which it is negative.
Determine the intervals in which the function $q(x) = -3x + 12$ is positive and those in which it is negative.
Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Find the axis intercepts and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Given the piecewise function: $f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$ Find its axis intercepts and study its sign…
Given the piecewise function:
$f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$
Find its axis intercepts and study its sign across its whole domain.
Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at the…
Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at the points $x = 4$ and $x = -4$.
Range of a function
State the range of the quadratic function $r(x) = x^2 + 4$.
State the range of the quadratic function $r(x) = x^2 + 4$.
Find the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.
Find the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.
Symmetry: even and odd functions
Study the symmetry (even, odd or neither) of the polynomial function $p(x) = x^4 - 2x^2 + 7$.
Study the symmetry (even, odd or neither) of the polynomial function $p(x) = x^4 - 2x^2 + 7$.
Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.
Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd or has no symmetry.
Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain.
Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain.
Increasing and decreasing behavior, extrema, and periodicity
And back to Yaritza's corner store: the profits rise to a maximum, fall, and if the pattern repeats every week, the function is periodic with a period of 7 days.
From the graphical representation of the parabola $y = f(x)$, state its intervals of increase and decrease, as well as the coordinates of its relative extremum.
From the graphical representation of the parabola $y = f(x)$, state its intervals of increase and decrease, as well as the coordinates of its relative extremum.
A projectile is launched vertically upward. The height $h(t)$ in meters that it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at…
A projectile is launched vertically upward. The height $h(t)$ in meters that it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at what time it reaches its maximum height and what this maximum height is.
A farmer has $100$ meters of fencing to enclose a rectangular vegetable garden taking advantage of an existing stone wall (no fencing is needed on that side). If $x$ is…
A farmer has $100$ meters of fencing to enclose a rectangular vegetable garden taking advantage of an existing stone wall (no fencing is needed on that side). If $x$ is the length of the sides perpendicular to the wall, the function describing the area is $A(x) = x(100-2x)$. Determine the domain of the function in the context of the problem and calculate the dimensions that maximize the area.
Observe the graph of the following periodic function modelling the voltage in an electronic circuit. Determine its period $T$ and calculate the value of $f(17)$.
Observe the graph of the following periodic function modelling the voltage in an electronic circuit. Determine its period $T$ and calculate the value of $f(17)$.
Determine the coefficients $a$, $b$ and $c$ of the quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(3, -4)$ and that it intersects the…
Determine the coefficients $a$, $b$ and $c$ of the quadratic function $f(x) = ax^2 + bx + c$ knowing that its vertex is the point $V(3, -4)$ and that it intersects the x-axis at $x = 1$.
The attached graph shows the height $h(t)$ in meters of a Ferris wheel capsule as a function of time $t$ in minutes. Given that the motion is modeled by a trigonometric…
The attached graph shows the height $h(t)$ in meters of a Ferris wheel capsule as a function of time $t$ in minutes. Given that the motion is modeled by a trigonometric function, determine its period, the maximum and minimum heights reached by the capsule, and the times at which they are reached during the first full cycle.
Asymptotes and complete analysis
An asymptote is a line that the graph gets closer and closer to without touching it (or without reaching it at that end). To find them for a rational function $f(x)=\dfrac{P(x)}{Q(x)}$:
To describe a graph, always follow this order: domain (which $x$ it covers), range (which $y$ it reaches), symmetry, asymptotes (vertical where the graph shoots off, horizontal at the sides), relative extrema, and sign (where it is above or below the $x$-axis). Take the average cost $C_m(x)=\dfrac{18{,}000x+72{,}000}{x}=18{,}000+\dfrac{72{,}000}{x}$ with $x>0$: it is always positive and decreasing, because if $x_2>x_1>0$ then $\frac{72{,}000}{x_2}<\frac{72{,}000}{x_1}$. It also has a vertical asymptote $x=0$ and a horizontal one $y=18{,}000$.
The average monthly cost (in pesos) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{18\,000x + 72\,000}{x}$, where $x$ is…
The average monthly cost (in pesos) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{18\,000x + 72\,000}{x}$, where $x$ is the number of units produced ($x > 0$). Determine the domain in the context of the problem, study its sign and prove analytically in which intervals the average cost is increasing or decreasing.
From the graph of the function $y = f(x)$, perform a complete analysis stating: domain, range, symmetry, equations of the asymptotes, relative extrema and intervals of…
From the graph of the function $y = f(x)$, perform a complete analysis stating: domain, range, symmetry, equations of the asymptotes, relative extrema and intervals of positive and negative sign.