Characteristics of Functions
Jayden's basketball coach models a free throw with $h(t) = -16t^2 + 20t + 6$, where $h$ is the ball's height in feet and $t$ is time in seconds. Before you can trust the model, you need answers to a few questions. When does it make sense? How high does the ball get, and when? When does it hit the floor? Answering questions like these is called analyzing a function. This page gives you a checklist: domain, intercepts, sign, increasing and decreasing behavior, extrema, range, symmetry, periodicity and asymptotes.
Domain: the three basic restrictions
The domain is the set of all inputs $x$ for which the function gives a real output. (Some problems say domain of definition; it means the same thing.) Start by assuming every real number works, then remove anything that breaks an operation.
| If $f$ contains… | Condition | Why |
|---|---|---|
| a fraction $\dfrac{\text{top}}{\text{bottom}}$ | bottom $\neq 0$ | you cannot divide by zero |
| an even root $\sqrt{\ \ }$ | radicand $\ge 0$ | no real square root of a negative number |
| a logarithm $\log_b(\ \ )$ or $\ln(\ \ )$ | argument $> 0$ | $\log_b u$ asks "which power of $b$ gives $u$?", and a positive base never gives $0$ or a negative |
Polynomials and odd roots (like $\sqrt[3]{x}$) have no restrictions: their domain is all real numbers.
A filled dot means the endpoint is included; an open dot means it is excluded.
Determine the domain of definition of the rational function $f(x) = \frac{x-1}{x^2-9}$.
Determine the domain of definition of the rational function $f(x) = \frac{x-1}{x^2-9}$.
Calculate the domain of the radical function $g(x) = \sqrt{2x-8}$.
Calculate the domain of the radical function $g(x) = \sqrt{2x-8}$.
Calculate the domain of the logarithmic function $s(x) = \log_3(x+2)$.
Calculate the domain of the logarithmic function $s(x) = \log_3(x+2)$.
Domain with several restrictions at once
Find the domain of definition of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.
Find the domain of definition of the function $f(x) = \sqrt{\frac{x+1}{x-3}}$.
Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x + 3)$.
Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x + 3)$.
Calculate the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, showing your reasoning.
Calculate the domain of the function $f(x) = \sqrt{\log_2(x-3)}$, showing your reasoning.
Intercepts
Why these rules? Every point on the y-axis has $x = 0$, and every point on the x-axis has $y = 0$. So you just plug in $0$ for the right variable.
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y-intercept: compute $f(0)$. If $0$ is not in the domain (like $\frac1x$), there is no y-intercept. A function has at most one.
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x-intercepts: solve $f(x) = 0$. For a fraction, a fraction is $0$ only when its numerator is $0$, so solve numerator $= 0$ and then check that the answers are in the domain.
Find the points of intersection with the coordinate axes of the polynomial function $h(x) = x^2 - 5x + 6$.
Find the points of intersection with the coordinate axes 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
A function can only change sign at a zero or at a point where it is undefined. So:
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Find the zeros of $f$.
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Find the values excluded from the domain.
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Mark them all on a number line. They split it into intervals.
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Test one number in each interval. The sign you find holds on the whole interval.
For a linear function you can skip the testing: $q(x) = 2x - 6$ is $0$ at $x = 3$, and since the slope is positive, $q < 0$ for $x < 3$ and $q > 0$ for $x > 3$.
Determine the intervals in which the function $q(x) = -3x + 12$ has a positive sign and in which it has a negative sign.
Determine the intervals in which the function $q(x) = -3x + 12$ has a positive sign and in which it has a negative sign.
Calculate the points of intersection with the axes and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Calculate the points of intersection with the axes and study the sign of the rational function $g(x) = \frac{x^2-4}{x-1}$.
Piecewise functions
A piecewise function uses a different formula on each part of its domain. To find intercepts and sign, work one piece at a time, and keep only the answers that lie inside that piece's interval.
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 points of intersection with the…
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 points of intersection with the axes and study its sign over its entire domain.
Increasing, decreasing and relative extrema
Intervals of increase and decrease are described with x-values (where along the x-axis), while extrema are points $(x, y)$. In the graph above, $f$ is decreasing on $(-\infty, 1)$, increasing on $(1, \infty)$, and has a relative minimum at $(1, -2)$.
Proving it with algebra. Take two numbers $x1 < x_2$ in the interval and look at the sign of $f(x_2) - f(x_1)$. If it is always positive, $f$ is increasing there; if always negative, decreasing. This is the same as checking that the average rate of change $\dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$ keeps one sign.
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.
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.
Parabolas: vertex, maximum and minimum
Many real situations (projectiles, areas, profits) are modeled by quadratics, and their best or worst value sits at the vertex.
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 $x = 1$.
Range
Ways to find it:
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From a graph: look at how low and how high the graph goes on the y-axis.
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For a parabola: the vertex is the lowest or highest value. If $a > 0$, the range is $[k, \infty)$; if $a < 0$, it is $(-\infty, k]$, where $k$ is the vertex's $y$-coordinate.
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With the inverse function: the range of $f$ is the domain of $f^{-1}$, since the inverse swaps the roles of $x$ and $y$.
State the range of the quadratic function $r(x) = x^2 + 4$.
State the range of the quadratic function $r(x) = x^2 + 4$.
Calculate the inverse function of $f(x) = \frac{2x-5}{x+1}$ and use it to determine the range of the function $f(x)$.
Calculate 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
How to test. First check that the domain is symmetric (if $x$ is in the domain, so is $-x$). Then replace $x$ by $-x$, simplify, and compare with $f(x)$ and with $-f(x)$.
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 of definition.
Prove analytically that the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$ is odd and determine its domain of definition.
Periodic functions
Some quantities repeat: a heartbeat, an AC voltage, the seat of a Ferris wheel.
To measure $T$ on a graph, pick any point and find the next point where the graph is at the same height and moving the same direction (for example, peak to peak). The horizontal distance between them is $T$.
Evaluating far away. Since $f(x + T) = f(x)$, you can subtract multiples of $T$ until you land inside the first cycle: $f(x) = f(x - nT)$. In practice, divide $x$ by $T$ and use the remainder.
Trigonometric models. A smooth wave like a Ferris wheel's height has three more features. The midline is halfway between the maximum and minimum, and the amplitude is half the distance between them:
$$\text{midline} = \frac{\max + \min}{2}, \qquad \text{amplitude} = \frac{\max - \min}{2}.$$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.
A complete analysis from a graph
When you analyze a graph, answer the same questions every time, in this order.
| Question | How to read it |
|---|---|
| Domain | Which $x$-values have a graph above or below them? Look for gaps and vertical asymptotes. |
| Range | Which $y$-values does the graph reach? |
| Symmetry | Mirror image across the y-axis (even) or the same after a $180^\circ$ turn (odd)? |
| Asymptotes | Vertical $x = a$: the graph shoots up or down near $x = a$. Horizontal $y = L$: the graph levels off toward $L$ as $x \to \pm\infty$. |
| Relative extrema | Peaks (maxima) and valleys (minima), as points $(x, y)$. |
| Sign | Where is the graph above the x-axis (positive) and below (negative)? |
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…
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.