Basic Concepts of Functions
At a gas station in Texas, regular costs $\$3.50$ per gallon. Pump 4 gallons and you pay $\$14$. Pump 10 gallons and you pay $\$35$. For every amount of gas there is exactly one price. That is a function. In this page you will answer two questions about any function: which inputs make sense, and how fast does the output change?
What Is a Function?
At the gas pump, $C(g) = 3.50g$ gives the cost in dollars for $g$ gallons. You choose $g$, so it is independent. The cost $C(g)$ is calculated, so it is dependent.
To evaluate a function, replace every $x$ with the input. Put the input in parentheses so negative numbers and exponents behave.
Writing a function from a situation
In a real situation the domain is limited by what makes sense. You cannot ride for a negative number of minutes, so the domain of $C$ is $t \ge 0$, or $[0, \infty)$.
The vertical line test
On a graph, a function never has two outputs for the same input. So a vertical line crosses its graph at most once.
Given the polynomial function $f(x) = 2x^2 - 5x + 1$, evaluate the function for the values $x = -2$, $x = 0$ and $x = 3$.
Given the polynomial function $f(x) = 2x^2 - 5x + 1$, evaluate the function for the values $x = -2$, $x = 0$ and $x = 3$.
A bicycle rental service charges a fixed fee of $\$3$ for registering for the service, plus $\$1,50$ for each hour of use. Write the algebraic expression of the function…
A bicycle rental service charges a fixed fee of $\$3$ for registering for the service, plus $\$1,50$ for each hour of use. Write the algebraic expression of the function that relates the total cost to the time of use, and identify which is the independent variable and which is the dependent variable.
A taxi company charges a $\$3$ base fee and $\$1,20$ for each mile traveled. a) Write the function that relates the total price of the ride to the miles traveled. b)…
A taxi company charges a $\$3$ base fee and $\$1,20$ for each mile traveled.
- a) Write the function that relates the total price of the ride to the miles traveled.
- b) What is the domain of this function taking into account the context of the problem?
- c) How much will a ride of $15$ miles cost?
- d) If a customer has paid $\$21$, how many miles have they traveled?
Finding the Domain
-
Polynomial (sums, differences, and products of $x$): defined for every $x$, so the domain is all real numbers, $(-\infty, \infty)$.
-
Fraction with $x$ in the denominator: exclude every $x$ that makes the denominator $0$ (you cannot divide by zero).
-
Even root ($\sqrt{\ }$, $\sqrt[4]{\ }$, …): the radicand must be $\ge 0$ (the square root of a negative number is not real).
Domains are written in interval notation. Use $\cup$ ("union") to join pieces: $[1, 5) \cup (5, \infty)$ means "all $x \ge 1$ except $5$".
A root of a fraction
When the radicand is a fraction, you need $\dfrac{\text{top}}{\text{bottom}} \ge 0$. Use a sign chart: find where the top and bottom are $0$, then test one number in each region.
Determine the domain of the following functions: a) $f(x) = \frac{3}{x - 5}$ b) $g(x) = \sqrt{x + 2}$
Determine the domain of the following functions:
- a) $f(x) = \frac{3}{x - 5}$
- b) $g(x) = \sqrt{x + 2}$
Find the domain of the function $f(x) = \frac{\sqrt{x + 4}}{x - 2}$.
Find the domain of the function $f(x) = \frac{\sqrt{x + 4}}{x - 2}$.
Determine algebraically the domain of the function $f(x) = \sqrt{\frac{x - 3}{x + 2}}$. Express the result as a union of intervals.
Determine algebraically the domain of the function $f(x) = \sqrt{\frac{x - 3}{x + 2}}$. Express the result as a union of intervals.
Intercepts and Symmetry
Why these names? For $f(x) = x^2$, the point $x$ and the point $-x$ give the same output, which is the mirror picture of the even powers. For $f(x) = x^3$, opposite inputs give opposite outputs, which is the picture of the odd powers.
Calculate analytically the x-intercepts and y-intercepts of the function $f(x) = x^2 - 6x + 8$.
Calculate analytically the x-intercepts and y-intercepts of the function $f(x) = x^2 - 6x + 8$.
Determine algebraically whether the following functions are even, odd or have no symmetry: a) $f(x) = x^4 - 2x^2$ b) $g(x) = x^3 - 3x + 1$
Determine algebraically whether the following functions are even, odd or have no symmetry:
- a) $f(x) = x^4 - 2x^2$
- b) $g(x) = x^3 - 3x + 1$
Average Rate of Change
How fast does the output change when the input changes? On the gas pump the answer was always $\$3.50$ per gallon. Most functions change at different speeds in different places, so we measure the change over an interval.
The units of the ARC are (output units) per (input unit): dollars per gallon, feet per second, miles per hour. For a linear function the ARC is the same on every interval, and it equals the slope.
Sometimes the ARC is given and an unknown is hidden in the function. Write the ARC formula with the unknown, set it equal to the given value, and solve. For example, if $f(x) = kx^2 - x$ has ARC $11$ on $[1, 3]$, then $\dfrac{(9k - 3) - (k - 1)}{2} = \dfrac{8k - 2}{2} = 4k - 1 = 11$, so $k = 3$.
Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4$ in the intervals $[-1, 2]$ and $[2, 5]$.
Calculate the Average Rate of Change (ARC) of the function $f(x) = x^2 - 4$ in the intervals $[-1, 2]$ and $[2, 5]$.
The distance $d$ (in feet) traveled by an object in free fall as a function of time $t$ (in seconds) is approximated by the function $d(t) = 5t^2$. Calculate the average…
The distance $d$ (in feet) traveled by an object in free fall as a function of time $t$ (in seconds) is approximated by the function $d(t) = 5t^2$. Calculate the average speed of the object (which corresponds to the Average Rate of Change) between the first and third second ($t \in [1, 3]$), and between the third and fifth second ($t \in [3, 5]$).
Given the quadratic function $f(x) = x^2 + kx$, determine the value of the parameter $k$ knowing that the Average Rate of Change of the function in the interval $[1, 3]$…
Given the quadratic function $f(x) = x^2 + kx$, determine the value of the parameter $k$ knowing that the Average Rate of Change of the function in the interval $[1, 3]$ is equal to $8$.
Reading a Graph: Domain, Range, Increasing and Decreasing
A graph shows everything at once, but you must read it in the right direction. Read from left to right.
-
Domain: all the $x$-values the graph covers (its "shadow" on the $x$-axis).
-
Range: all the $y$-values the graph reaches (its "shadow" on the $y$-axis).
-
A filled dot means the point is part of the graph. An open circle means it is not. An arrow means the graph continues forever.
-
Increasing on an interval: the graph rises as $x$ increases. Decreasing: the graph falls.
-
A relative maximum is a peak, higher than the points right around it. A relative minimum is a valley. Report them as points $(x, y)$.
Intervals of increase and decrease are always given in terms of $x$.
Connection to the ARC. If $f$ is increasing on all of $[a, b]$, then the ARC on $[a, b]$ is positive. If $f$ is decreasing on all of $[a, b]$, the ARC is negative. For example, a negative ARC on an interval where the graph is falling confirms that the function is decreasing there.
From the graph of the function $h(x)$, indicate its domain and its range in interval notation.
From the graph of the function $h(x)$, indicate its domain and its range in interval notation.
Observe the following graph of a function $f(x)$ and determine its intervals of increase and decrease, as well as the exact coordinates of its relative maxima and minima.
Observe the following graph of a function $f(x)$ and determine its intervals of increase and decrease, as well as the exact coordinates of its relative maxima and minima.
Calculate the Average Rate of Change of the function $f(x) = \frac{1}{x}$ in the intervals $[1, 2]$ and $[2, 4]$. What does the sign of the results indicate about the…
Calculate the Average Rate of Change of the function $f(x) = \frac{1}{x}$ in the intervals $[1, 2]$ and $[2, 4]$. What does the sign of the results indicate about the monotonicity of the function in those intervals?
Maximum and Minimum Values
Parabolas
For $f(x) = ax^2 + bx + c$, the highest or lowest point is the vertex.
Why does this work? The parabola is symmetric about the vertical line $x = -\frac{b}{2a}$, and the vertex sits on that line of symmetry.
Optimization with a function
Now change the situation: what if one side of the pen is an existing wall? Then you only fence three sides, and the constraint changes.
Periodic graphs
Some graphs repeat the same pattern again and again. A function is periodic if its graph repeats at regular intervals. The period is the length of one full cycle, measured between two matching points such as two consecutive peaks. A periodic function satisfies $f(t + \text{period}) = f(t)$.
The attached graph represents the height in feet of a Ferris wheel cabin as a function of time in minutes. Observe the graph and answer: a) Is it a periodic function? If…
The attached graph represents the height in feet of a Ferris wheel cabin as a function of time in minutes. Observe the graph and answer:
- a) Is it a periodic function? If so, indicate its period.
- b) What is the maximum and minimum height reached by the cabin?
- c) At what height is the cabin exactly $10$ minutes after it starts turning?
The temperature $T$ (in degrees Fahrenheit) inside an industrial oven from when it is turned on until it is turned off is given by the function $T(t) = -t^2 + 8t + 20$,…
The temperature $T$ (in degrees Fahrenheit) inside an industrial oven from when it is turned on until it is turned off is given by the function $T(t) = -t^2 + 8t + 20$, where $t$ is the time elapsed in hours. At what instant is the maximum temperature reached? What is the value of that maximum temperature?
A farmer has $100$ feet of fencing to enclose a rectangular garden plot, making use of the straight wall of an already built warehouse, so that he only needs to fence…
A farmer has $100$ feet of fencing to enclose a rectangular garden plot, making use of the straight wall of an already built warehouse, so that he only needs to fence three of the four sides of the rectangle.
- a) Find the expression of the function that gives the area of the garden plot as a function of the length $x$ of one of the sides perpendicular to the wall.
- b) Determine the valid domain for $x$ in the context of this problem.
- c) What dimensions must the garden plot have for its area to be maximum?
Asymptotes
A rational function is a fraction of two polynomials. Its graph can get closer and closer to a line without touching it. That line is an asymptote.
When the denominator is a polynomial of degree $2$ or more, factor it to find all of its zeros. Each zero that does not also cancel with the numerator gives a vertical asymptote.
Determine algebraically the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{2x + 1}{x - 3}$.
Determine algebraically the equations of the vertical and horizontal asymptotes of the rational function $f(x) = \frac{2x + 1}{x - 3}$.
Given the rational function $f(x) = \frac{ax^2 + 3}{2x^2 - x}$, determine the value of the parameter $a$ so that the function has a horizontal asymptote at the line…
Given the rational function $f(x) = \frac{ax^2 + 3}{2x^2 - x}$, determine the value of the parameter $a$ so that the function has a horizontal asymptote at the line $y = 4$. With that value of $a$ already calculated, also find the equations of the vertical asymptotes of the function.
Given the rational function $f(x) = \frac{x^2}{x^2 - 4}$: a) Prove algebraically whether it is an even function, an odd function or if it has no symmetry. b) Calculate…
Given the rational function $f(x) = \frac{x^2}{x^2 - 4}$:
- a) Prove algebraically whether it is an even function, an odd function or if it has no symmetry.
- b) Calculate the x- and y-intercepts.
- c) Find the equations of all its asymptotes (vertical and horizontal).
Piecewise Functions and Continuity
A function is continuous at a point where the pieces meet if you can draw through it without lifting your pencil. For piecewise formulas, that means both rules give the same output at the boundary.
If a function has two boundaries and two unknown parameters, treat each boundary separately. Each boundary gives its own equation.
Types of discontinuity
When a graph is not continuous at a point, it breaks in one of three ways.
-
Jump: the two sides approach different heights.
-
Hole (removable): both sides approach the same height, but the point itself is missing or sits somewhere else.
-
Infinite: the graph shoots up or down along a vertical asymptote.
When you read a value from a graph of a piecewise function, go to the $x$ on the horizontal axis and move up or down to the graph. Use the filled dot. An open circle is not part of the graph.
Step functions
Prices like "$\$4$ for the first pound or part of a pound" make step functions. "Or part of" means you round up to the next whole unit. Be careful with endpoints.
In the graph below, an open circle marks an endpoint that is not included (left ends of each step), and a filled dot marks an endpoint that is included (right ends).
Given the following graph of a piecewise function, find the values of $f(-2)$, $f(1)$ and $f(4)$. Then, indicate at which points the function is discontinuous and what…
Given the following graph of a piecewise function, find the values of $f(-2)$, $f(1)$ and $f(4)$. Then, indicate at which points the function is discontinuous and what type each discontinuity is.
Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous throughout its domain:…
Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous throughout its domain:
$f(x) = \begin{cases} 2x + a & \text{if } x < 1 \\ x^2 - 3x + 5 & \text{if } 1 \le x < 4 \\ bx - 1 & \text{if } x \ge 4 \end{cases}$
A public parking lot has the following rate: it charges $\$2$ for the first hour (or fraction of an hour), and $\$1,50$ for each additional hour (or fraction). The…
A public parking lot has the following rate: it charges $\$2$ for the first hour (or fraction of an hour), and $\$1,50$ for each additional hour (or fraction). The maximum cost per day is $\$12$. Write the algebraic expression of the piecewise function that relates the cost to the parking time $t$ (in hours) for the first $5$ hours, clearly indicating the domain intervals for each section.
Putting It All Together
To analyze a function, go through this checklist. Some items you compute from the formula. Others you read from the graph.
| Question | How to answer it |
|---|---|
| Domain | Exclude zeros of denominators, require radicands $\ge 0$ |
| Range | Read from the graph (lowest and highest $y$) |
| Intercepts | $f(0)$ for the y-intercept, $f(x) = 0$ for x-intercepts |
| Symmetry | Compare $f(-x)$ with $f(x)$ and $-f(x)$ |
| Asymptotes | Zeros of the denominator, degrees for the horizontal one |
| Increasing / decreasing | Read the graph left to right (in terms of $x$) |
| Relative extrema | Peaks and valleys, written as points $(x, y)$ |
Analyze the following graph of the function $f(x)$ and determine: a) Its domain and range. b) The equations of its asymptotes. c) The intervals of increase and decrease.…
Analyze the following graph of the function $f(x)$ and determine:
- a) Its domain and range.
- b) The equations of its asymptotes.
- c) The intervals of increase and decrease.
- d) The coordinates of its relative extrema (maxima and minima).
Summary
Back to the gas pump: $C(g) = 3.50g$ makes sense for $g \ge 0$, up to the size of the tank. Its average rate of change on any interval is $\$3.50$ per gallon, which is its slope, the unit price.