Studying a Function
Ximena graphs on her computer the attendance at a movie theater in Mexico City during the weeks after a premiere. How does she know where it starts, when it rises, when it falls, and what value it levels off at, without calculating a hundred points? By always asking the same questions in the same order. That is called "studying a function," and by the end of this page you will do it yourself.
The outline for studying a function
Domain and range
The domain is the set of $x$-values where there is a graph; the range is the set of $y$-values it reaches. On a graph, the domain is read by projecting onto the $X$-axis and the range by projecting onto the $Y$-axis. A point with a closed circle is included (bracket); one with an open circle is not (parenthesis).
For a formula, look for what the calculation forbids:
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A denominator cannot be $0$.
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A square root requires a radicand $\ge 0$.
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A polynomial forbids nothing: its domain is $\mathbb{R}$.
Find the domain of definition of the irrational function $f(x) = \sqrt{2x - 8}$.
Find the domain of definition of the irrational function $f(x) = \sqrt{2x - 8}$.
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…
Based on a function whose graph consists of two continuous sections:
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The first section goes from the point $(-3, 2)$ (included) to the point $(1, 4)$ (not included), and has a relative maximum at $(-1, 5)$.
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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.
Find the domain of definition of the rational function $f(x) = \frac{x - 1}{x^2 - 4x - 5}$.
Find the domain of definition of the rational function $f(x) = \frac{x - 1}{x^2 - 4x - 5}$.
Intercepts and symmetry
Find the axis intercepts of the function $f(x) = \frac{3x - 6}{x + 2}$.
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.
Determine analytically whether the function $f(x) = x^4 - 5x^2 + 3$ has even symmetry, odd symmetry or no symmetry.
Study the symmetry of the function $f(x) = \frac{x}{x^2 + 1}$ and calculate its axis intercepts.
Study the symmetry of the function $f(x) = \frac{x}{x^2 + 1}$ and calculate its axis intercepts.
Monotonicity, extrema, and average rate of change
A function is increasing on an interval if, moving to the right, the graph goes up; decreasing if it goes down; constant if it stays at the same height. A relative maximum is a point where the function goes from increasing to decreasing; a relative minimum, where it goes from decreasing to increasing.
Continuity at a point
A function is continuous at $x=a$ if its graph does not break there: the value of $f(a)$ matches the value that the two formulas approach as they get close to $a$ from the left and from the right. In a piecewise function, you have to compare the two formulas at the joining point.
Calculate the average rate of change (ARC) of the function $f(x) = 2x^2 - 3x$ on the interval $[-1, 2]$.
Calculate the average rate of change (ARC) of the function $f(x) = 2x^2 - 3x$ on the interval $[-1, 2]$.
From the following graph, state the intervals of monotonicity (increasing and decreasing) and the coordinates of its local extrema.
From the following graph, state the intervals of monotonicity (increasing and decreasing) and the coordinates of its local extrema.
Study the continuity of the following piecewise function at the point $x = 1$: $f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$
Study the continuity of the following piecewise function at the point $x = 1$:
$f(x) = \begin{cases} 2x + 1 & \text{if } x \leq 1 \\ 4 - x & \text{if } x > 1 \end{cases}$
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…
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.
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 meters and $t$ is the time in seconds.…
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 meters 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.
Absolute value and piecewise functions
Absolute value turns whatever is inside into a positive number: $|u|=u$ if $u\ge 0$ and $|u|=-u$ if $u<0$. To study $|{\text{expression}}|$, first find where the expression is positive or negative and write a formula for each case.
Concavity
A graph is concave up (also called convex) if it curves like a "U," and concave down if it curves like a "∩." For a parabola $y=ax^2+bx+c$ this is decided by the sign of $a$: $a>0$ concave up, $a<0$ concave down.
Express the modulus function $f(x) = |2x - 4|$ as a piecewise function and determine its increasing and decreasing intervals.
Express the modulus function $f(x) = |2x - 4|$ as a piecewise function and determine its increasing and decreasing intervals.
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…
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 $g(x)$.
Given the following piecewise function, study its domain, its continuity, its axis intercepts and its intervals of monotonicity:…
Given the following piecewise function, study its domain, its continuity, its axis intercepts and its intervals of monotonicity:
$f(x) = \begin{cases} x^2 - 4 & \text{if } x < 0 \\ 2x - 4 & \text{if } 0 \leq x < 4 \\ 4 & \text{if } x \geq 4 \end{cases}$
Models with parabolas: the vertex as an extremum
A quadratic function $f(x)=ax^2+bx+c$ has a single extremum, its vertex, at
$$x_v=-\frac{b}{2a}$$If $a<0$ it is a maximum; if $a>0$, a minimum. Why? The parabola is symmetric about the vertical line through its highest or lowest point, which lies exactly halfway between its two intercepts with the $X$-axis.
Expressing an area as a function
To optimize, you first have to write what you want as a function of a single variable. Geometric relationships are used, such as similarity of triangles.
The variation in the water level of a reservoir $v$ (in meters relative to its average level) as a function of the month $x$ of the year is given by the function…
The variation in the water level of a reservoir $v$ (in meters 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 ($v = 0$).
A projectile is launched upward and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in meters and $t$ is the time in…
A projectile is launched upward and follows a trajectory described by the function $h(t) = -5t^2 + 20t + 25$, where $h$ is the height in meters 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.
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…
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.
End behavior and asymptotes
For a fraction of polynomials, compare the degrees of the numerator and the denominator:
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Numerator of lower degree: the function tends to $0$; asymptote $y=0$.
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Equal degrees: it tends to the quotient of the leading coefficients.
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Numerator of higher degree: it grows without approaching a value; there is no horizontal asymptote.
Finding parameters from the asymptotes
In $f(x)=\dfrac{ax+b}{x+c}$, the vertical asymptote is $x=-c$ and the horizontal one is $y=a$. With one more point you get $b$.
Identify the equations of the vertical asymptote and the horizontal asymptote of the function $f(x) = \frac{5}{x - 3} + 2$.
Identify the equations of the vertical asymptote and the horizontal asymptote of the function $f(x) = \frac{5}{x - 3} + 2$.
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{3x + 1}{x - 2}$.
Determine the equations of the vertical and horizontal asymptotes of the function $f(x) = \frac{3x + 1}{x - 2}$.
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…
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…
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 $t \to \infty$).
The complete profile of a function
To sketch a graph from given conditions, first draw the asymptotes as dotted lines, then the given points (intercepts and extrema), apply the symmetry, and finally connect them with strokes that respect increasing and decreasing behavior.
Back to the beginning: Ximena, with the outline domain → intercepts → symmetry → monotonicity → asymptotes, describes the theater attendance with just a few data points and doesn't need to tabulate a hundred points.
Describe the behavior 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…
Describe the behavior 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.
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…
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.
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…
Sketch the graph of a function $f(x)$ in your notebook that satisfies all of the following conditions simultaneously:
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Its domain is $\mathbb{R} \setminus \{-2, 2\}$.
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It is a function with odd symmetry.
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It has vertical asymptotes at $x = -2$ and $x = 2$, and a horizontal asymptote at $y = 0$.
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It has a local maximum at the point $(-4, 1)$ and a local minimum at $(4, -1)$.
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It passes through the origin.
You can do it on paper and send a photo with your answer.