Properties of Functions
Your friend is on the phone and can't see your Desmos screen. "What does the graph look like?" Saying "kind of a hill" won't cut it. To describe a function exactly, you answer the same questions every time: Where is it defined? Where does it cross the axes? Where is it positive or negative? Which values does it reach? Is it symmetric? Does it rise or fall? Does it repeat? In this page you learn to answer all of them, from a formula and from a graph.
Domain: one restriction at a time
The domain of $f$ is the set of all inputs $x$ for which $f(x)$ makes sense. Unless a problem says otherwise, it is every real number except those that "break" the formula. Only three things can break a formula at this level.
| Danger | Rule | Example |
|---|---|---|
| Division | denominator $\ne 0$ | $\dfrac{1}{x-5}$ needs $x \ne 5$ |
| Even root (square root, 4th root, …) | radicand $\ge 0$ | $\sqrt{x-5}$ needs $x \ge 5$ |
| Logarithm | argument $> 0$ (strictly) | $\ln(x-5)$ needs $x > 5$ |
Why? You can't divide by zero, you can't take a real square root of a negative number, and no power of a positive base ever equals $0$ or a negative number, so $\log$ of those doesn't exist. Polynomials and odd roots (such as $\sqrt[3]{x}$) never break: their domain is all real numbers.
Domains are written in interval notation. Use $\cup$ ("union") to join pieces, for example $(-\infty, 2) \cup (2, \infty)$, which can also be written $\mathbb{R} \setminus \{2\}$ ("all real numbers except 2").
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 $s(x) = \log_3(x+2)$.
Domain: several restrictions at once
When a formula mixes a fraction, a root and a logarithm, each piece sets its own condition. A number is in the domain only if all conditions hold at the same time, so the domain is the intersection of the conditions.
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 $f(x) = \sqrt{\log_2(x-3)}$, showing your reasoning.
Find the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, showing your reasoning.
Intercepts
To find zeros of a polynomial, factor it or use the quadratic formula. To find zeros of a fraction, set the numerator equal to $0$, then discard any value that also makes the denominator $0$.
Look at the graph above: the curve crosses the x-axis at $-4$ and $4$ and the y-axis at $-8$. It also breaks at $x = -2$, the number that is not in the domain.
Find the x- and y-intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.
Find the x- and y-intercepts of the polynomial function $h(x) = x^2 - 5x + 6$.
Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has the domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at…
Find the values of the parameters $a$ and $b$ so that the function $f(x) = \frac{x^2+a}{x-b}$ has the domain $\mathbb{R} \setminus \{2\}$ and intersects the x-axis at the points $x = 4$ and $x = -4$.
Sign of a function
The sign of $f$ tells you where the graph is above the x-axis ($f(x) > 0$, positive) and where it is below ($f(x) < 0$, negative).
Why does a method work? A function can change sign only at a zero, or where it is undefined. Between two such consecutive numbers the sign cannot change, so one test value tells you the sign of the whole interval.
For a linear function the work is short: $q(x) = 2x - 6$ is $0$ at $x = 3$, negative to the left and positive to the right, because the slope is positive.
Piecewise functions. Treat each piece only on its own interval, and discard any solution outside it.
Determine the intervals where the function $q(x) = -3x + 12$ is positive and where it is negative.
Determine the intervals where the function $q(x) = -3x + 12$ is positive and where it is negative.
Find the x- and y-intercepts and determine the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Find the x- and y-intercepts and determine the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Given the piecewise-defined function: $f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$ Find its x- and y-intercepts and…
Given the piecewise-defined function:
$f(x) = \begin{cases} 2x+6 & \text{if } x < -1 \\ x^2+3 & \text{if } x \ge -1 \end{cases}$
Find its x- and y-intercepts and determine its sign over its entire domain.
Range
There are two reliable ways to find it.
Method 1: use the shape. A parabola $y = a(x-h)^2 + k$ has its vertex at $(h, k)$. If $a > 0$ the range is $[k, \infty)$. If $a < 0$ it is $(-\infty, k]$.
Method 2: solve for $x$. A number $y$ is in the range exactly when the equation $y = f(x)$ has a solution $x$. Solving for $x$ gives the inverse function, and the range of $f$ is the domain of the inverse.
Find the range of the quadratic function $r(x) = x^2 + 4$.
Find 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
Why test $f(-x)$? The point $(-x, f(-x))$ is the mirror image of $(x, f(x))$ across the y-axis. If the heights match, the graph is a mirror image. If they are opposites, the graph turns into itself under a half-turn.
Determine the symmetry (even, odd, or neither) of the polynomial function $p(x) = x^4 - 2x^2 + 7$.
Determine 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 neither.
Determine analytically whether the function $h(x) = \frac{x^3}{x^2+1}$ is even, odd, or neither.
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, decreasing and extrema
Read a graph from left to right. If it climbs, the function is increasing. If it falls, it is decreasing. Formally, on an interval $I$:
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$f$ is increasing if $x1 < x_2$ implies $f(x_1) < f(x_2)$.
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$f$ is decreasing if $x1 < x_2$ implies $f(x_1) > f(x2)$.
The intervals are intervals of $x$. A point where the graph turns from decreasing to increasing is a relative minimum. From increasing to decreasing it is a relative maximum. Both are called relative extrema, and you report them as a point $(x, y)$.
From this graph: decreasing on $(-\infty, 1)$, increasing on $(1, \infty)$, relative minimum at $(1, -2)$. For a parabola $y = ax^2 + bx + c$ the turning point is the vertex, at $x = -\dfrac{b}{2a}$.
Proving it with algebra. Take two numbers $x1 < x_2$ in the interval and study the sign of $f(x_2) - f(x_1)$. If it is always positive the function is increasing, and if it is always negative it is decreasing.
From the graph of the parabola $y = f(x)$, find its intervals of increase and decrease, as well as the coordinates of its relative extremum.
From the graph of the parabola $y = f(x)$, find its intervals of increase and decrease, as well as the coordinates of its relative extremum.
The average monthly cost (in dollars) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{300x + 1200}{x}$, where $x$ is the…
The average monthly cost (in dollars) per unit produced in an electronic components factory is given by the function $C_m(x) = \frac{300x + 1200}{x}$, 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.
Quadratic models: maximum and minimum
Many real situations are modeled by a parabola, and the question is almost always "what is the highest (or lowest) value?" The answer is the vertex.
Move the sliders: $h$ and $k$ slide the vertex, and the sign of $a$ flips the parabola between a hill and a valley.
A projectile is launched vertically upward. The height $h(t)$ in feet it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at what…
A projectile is launched vertically upward. The height $h(t)$ in feet it reaches after $t$ seconds is given by the function $h(t) = -5t^2 + 20t + 1$. Determine at what instant it reaches the maximum height and what the value of this height is.
A farmer has $100$ feet of fencing to enclose a rectangular vegetable patch, taking advantage of an existing stone wall (no fence is needed on that side). If $x$ is the…
A farmer has $100$ feet of fencing to enclose a rectangular vegetable patch, taking advantage of an existing stone wall (no fence is needed on that side). If $x$ is the length of the sides perpendicular to the wall, the function that describes 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.
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 $x = 1$.
Periodic functions
A periodic function repeats its values at regular intervals: there is a number $T > 0$ with $f(x + T) = f(x)$ for all $x$. The smallest such $T$ is the period. Ferris wheels, sound waves, tides and the AC voltage in an outlet are all periodic.
Reading the period from a graph: measure the horizontal distance between two consecutive identical points, for example peak to peak or valley to valley. Peak to valley is only half a period.
For a wave that oscillates smoothly (sinusoidal), you also read:
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Maximum $M$ and minimum $m$ heights.
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Midline: $\dfrac{M + m}{2}$.
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Amplitude: $\dfrac{M - m}{2}$, the distance from the midline to a peak.
Evaluating far away. If you know one full period, you can find any value by subtracting multiples of $T$ until you land inside the known cycle.
Observe the graph of the following periodic function that models 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 that models the voltage in an electronic circuit. Determine its period $T$ and calculate the value of $f(17)$.
The attached graph shows the height $h(t)$ in feet of a Ferris wheel cabin as a function of time $t$ in minutes. Knowing that the movement is modeled by a trigonometric…
The attached graph shows the height $h(t)$ in feet of a Ferris wheel cabin as a function of time $t$ in minutes. Knowing that the movement is modeled by a trigonometric function, determine its period, the maximum and minimum heights reached by the cabin, and the instants of time at which they are reached during the first complete cycle.
Putting it together: a full analysis from a graph
Now you can describe any function. Here is where to look on a graph.
| Property | How to read it from the graph |
|---|---|
| Domain | the $x$-values where the graph exists; holes and vertical asymptotes are excluded |
| Range | the $y$-values the graph reaches |
| Symmetry | mirror image across the y-axis: even. Half-turn about the origin: odd. Otherwise neither |
| Intercepts | where the graph meets the axes |
| Sign | above the x-axis: positive; below: negative |
| Asymptotes | dashed lines the graph approaches: a vertical asymptote $x = a$ where the graph shoots up or down, a horizontal asymptote $y = b$ where the ends level off |
| Increasing / decreasing | read left to right |
| Extrema | peaks and valleys, as points $(x, y)$ |
From the graph of the function $y = f(x)$, carry out a complete analysis indicating: domain, range, symmetry, equations of the asymptotes, relative extrema, and…
From the graph of the function $y = f(x)$, carry out a complete analysis indicating: domain, range, symmetry, equations of the asymptotes, relative extrema, and intervals of positive and negative sign.
Back to the start: you now have a checklist for describing any graph over the phone. Domain, range, intercepts, sign, symmetry, increasing and decreasing, extrema, asymptotes, and period if it repeats.