Functions and Their Graphs

One input, one out­put#

Put a coin in a vend­ing machine, press a but­ton, and exactly one thing comes out. A func­tion behaves the same way: it is a rela­tion in which every ele­ment of the domain has exactly one image. Func­tions are the back­bone of cal­cu­lus, so this les­son mat­ters. We will test rela­tions to see whether they are func­tions, get to know the stan­dard real func­tions (iden­tity, con­stant, poly­no­mial, ratio­nal, mod­u­lus, signum and great­est inte­ger) through their graphs, domains and ranges, and then add, sub­tract, mul­ti­ply and divide func­tions.

Func­tions#

A rela­tion $f$ from $A$ to $B$ is a func­tion if every $a \in A$ appears as a first ele­ment exactly once: not zero times, not twice. We write $f : A \to B$ and $y = f(x)$.

Ver­ti­cal line test. With a graph in front of you there is a quick check. A curve in the plane is the graph of a func­tion exactly when no ver­ti­cal line meets it twice. A cir­cle fails this test.

Real func­tions#

The great­est inte­ger func­tion needs a word of care. $[x]$ is the great­est inte­ger not exceed­ing $x$, so $[2.7] = 2$ and $[5] = 5$. With neg­a­tives stu­dents often slip: $[-2.7] = -3$, because you must step down to the inte­ger just below, not up towards zero. Its graph looks like a stair­case.

A poly­no­mial func­tion has the form $f(x) = a_0 + a_1 x + \cdots + a_n x^n$, where $n$ is a whole num­ber. That rules out $\sqrt{x}$ and $\displaystyle \tfrac{1}{x}$, which are not poly­no­mi­als. A ratio­nal func­tion is a quo­tient $\displaystyle \tfrac{p(x)}{q(x)}$ of poly­no­mi­als, defined wher­ever $q(x) \ne 0$.

For the domain, ask what would break the for­mula: a zero denom­i­na­tor, or a neg­a­tive num­ber under a square root. For the range, ask what val­ues the out­put can actu­ally reach.

Find­ing domain and range#

Alge­bra of func­tions#

Func­tions can be com­bined point by point, just like num­bers. For $f, g : X \to \mathbf{R}$ we define $(f + g)(x) = f(x) + g(x)$, $(f - g)(x) = f(x) - g(x)$, $(fg)(x) = f(x) g(x)$ and $(\alpha f)(x) = \alpha f(x)$. For the quo­tient, $\displaystyle \left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)}$, but only where $g(x) \ne 0$.

Prac­tice ques­tions#

1. Which rela­tions on $\{1, 2, 3, 4\}$ are func­tions? (a) $\{(1, 3), (2, 3), (3, 3), (4, 3)\}$; (b) $\{(1, 1), (2, 1), (2, 4), (3, 2), (4, 3)\}$; (c) $\{(1, 4), (2, 3), (3, 2)\}$.
2. $f(x) = x^2 - 4x$. Find $f(0)$, $f(-1)$, $f(4)$, $f(a + 1)$.
3. Eval­u­ate: $[3.9]$; $[-0.4]$; $[-6]$; $\lvert -7 \rvert - \lvert 2 \rvert$.
4. Find the domain: $\displaystyle \frac{1}{x^2 - 9}$; $\sqrt{8 - 2x}$; $\displaystyle \frac{x}{x^2 + 4}$.
5. Find the range: $x^2 + 5$; $\lvert x + 1 \rvert - 3$; $\displaystyle \frac{1}{x^2 + 1}$.
6. $f(x) = 2x - 3$ and $g(x) = x^2$. Find $f + g$, $fg$, $\displaystyle \tfrac{g}{f}$ and state the domain of $\displaystyle \tfrac{g}{f}$.
7. Sketch $f(x) = \lvert x - 2 \rvert$ and $f(x) = [x]$ for $-2 \le x < 3$.
8. $f = \{(x, 3x - 1) : x \in \mathbf{Z}\}$. Find $f(4)$ and the $x$ with $f(x) = 14$.
9. Is $\{(x, y) : y^2 = x\}$ a func­tion of $x$ on $[0, \infty)$? Why?
10. A taxi charges ₹$40$ plus ₹$15$ per km. Write the fare as a func­tion of dis­tance and find the fare for $12$ km.

Answers#