Some Functions and Their Graphs

The Graph of the Identity Function

The figure shows a standard Cartesian plane with both axes drawn as double-arrowed lines, extending from approximately −8 to 8 on each axis. Even-numbered tick marks are labelled along both axes. A single straight line, coloured indigo, runs through the origin at a 45° angle — it has slope exactly 1. The line has arrowheads at both ends, indicating it continues indefinitely in both directions.

Every point on this line satisfies the condition that its $x$-coordinate equals its $y$-coordinate. The line passes through $(-2,-2)$, $(-1,-1)$, $(0,0)$, $(1,1)$, $(2,2)$, and so on. There are no breaks, no curves, no other lines — just this one diagonal.

What This Figure Teaches

The identity function $f(x) = x$ is the simplest possible function that still does something: it takes each input and returns it unchanged. The graph makes this idea visually obvious. If you pick any $x$ on the horizontal axis, the corresponding $y$ on the vertical axis is exactly the same number. The line $y = x$ is the set of all points where the two coordinates are equal.

This is the baseline against which all other functions are compared. When you later study transformations — shifting, stretching, reflecting — the identity function is the starting point. A vertical shift upward by 2 units, for instance, turns $y = x$ into $y = x + 2$. A reflection across the $x$-axis turns it into $y = -x$. Understanding the identity function's graph first makes those modifications easy to see.

The Key Formula

The central relationship the figure illustrates is:

Here:

• $f$ is the function name
• $x$ is the input variable (any real number)
• $f(x)$ is the output, which equals the input
• $\mathbb{R}$ denotes the set of all real numbers …

The Graph of a Constant Function

Figure 2.9 shows the simplest possible non-trivial function: a constant function. The axes are the standard Cartesian frame — a horizontal $x$-axis and a vertical $y$-axis, meeting at the origin. Drawn across this plane is a single horizontal line at height $y = 3$, running parallel to the $x$-axis. The line has arrowheads on both ends, indicating that it extends infinitely in both directions along the $x$-axis.

The physical idea is straightforward: no matter what $x$-value you choose, the output $y$ is always the same number. In this case, $f(x) = 3$ for every real $x$. The graph is a horizontal line because the $y$-coordinate never changes — it is constant. The line is perfectly flat, with zero slope.

Here $c$ is a fixed real number (the constant), and $\mathbb{R}$ denotes the set of all real numbers. The domain of this function is $\mathbb{R}$ (every real number is allowed as input), and the range is the single-element set $\{c\}$ — only one output value ever appears.

The line in the figure is at $y = 3$, so $c = 3$ for this example. Every point on that line has coordinates of the form $(x, 3)$, where $x$ can be any real number. The line is parallel to the $x$-axis because the $y$-coordinate is fixed; it never rises or falls as $x$ changes.

The Graph of $f(x) = x^2$

Figure 2.10 shows the graph of the quadratic function $f(x) = x^2$, defined for all real numbers $x$. The axes are drawn in the standard Cartesian plane: the horizontal $x$-axis and vertical $y$-axis intersect at the origin $(0,0)$. The curve itself is an upward-opening parabola, coloured indigo in the textbook.

The parabola is symmetric about the $y$-axis. This means the left half ($x < 0$) is a mirror image of the right half ($x > 0$). The vertex — the lowest point on the curve — sits exactly at the origin. From there, the graph rises smoothly in both directions, becoming steeper as $|x|$ increases.

The table in Example 13 gives the key points plotted: $(-4,16)$, $(-3,9)$, $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,4)$, $(3,9)$, $(4,16)$. These points lie on the parabola and confirm its shape. The curve passes through every such point, forming a continuous, unbroken line.

What the Figure Teaches

The graph makes three ideas immediately visible.

First, domain and range. The domain of $f(x) = x^2$ is all real numbers $\mathbb{R}$ — you can square any real $x$. The range, however, is only non-negative real numbers: $[0, \infty)$. The graph stays on or above the $x$-axis, showing that $y$ never takes a negative value.

Second, even symmetry. For any $x$, $f(-x) = (-x)^2 = x^2 = f(x)$. This is why the curve is symmetric about the $y$-axis. Functions with this property are called even functions.

Third, the shape of a quadratic. The parabola is the simplest non-linear polynomial graph. Its U-shape, with a single minimum at the vertex, is the foundation for understanding all quadratic functions $ax^2 + bx + c$.

The Key Formula

The function graphed is

where:

• $x$ is any real number (the input, or independent variable) …

The figure shows the graph of the function $f(x) = x^{3}$ drawn on a standard Cartesian plane. The horizontal axis is the $x$-axis (the domain, all real numbers), and the vertical axis is the $y$-axis (the range, also all real numbers). The curve itself is an S-shaped cubic that passes through the origin $(0,0)$. Near the origin the graph is very flat — it hugs the $x$-axis for a short stretch — but as $x$ moves away from zero in either direction, the curve steepens dramatically and rises (or falls) without bound.

The key idea the figure teaches is that the cubic function $f(x) = x^{3}$ is an odd function: it is symmetric about the origin. For every point $(a, a^{3})$ on the graph, the point $(-a, -a^{3})$ is also on the graph. This is why the curve goes through the origin and why the left-hand side is a mirror image of the right-hand side, but flipped upside down. The flatness near $x = 0$ reflects the fact that for small $x$, $x^{3}$ is much smaller than $x$ — for example, $0.1^{3} = 0.001$ — so the graph barely rises. The steepness away from the origin shows that $x^{3}$ grows faster than any linear or quadratic function for large $|x|$.

The central formula the textbook develops with this figure is simply the definition of the function itself:

Here $f$ is a real-valued function from $\mathbb{R}$ to $\mathbb{R}$. The symbol $x$ represents any real number (the input), and $x^{3}$ is the output. The domain is all real numbers, and the range is also all real numbers — every real number has a real cube root, so the graph covers the entire vertical axis.

The textbook uses this figure as part of a sequence introducing basic polynomial functions. After the identity function $f(x) = x$ (a straight line) and the square function $f(x) = x^{2}$ (a parabola), the cube function $f(x) = x^{3}$ is the next natural example. The figure makes the contrast clear: unlike the parabola (which is even and U-shaped), the cubic is odd and S-shaped, and unlike the identity line, it is not straight. …

Fig. 2.12 shows the graph of the function $f(x) = \frac{1}{x}$, defined for all real numbers except $x = 0$. The axes are drawn in the usual way: a horizontal $x$-axis and a vertical $y$-axis, crossing at the origin. The curve consists of two separate branches, each drawn in indigo. One branch lies entirely in the first quadrant (where $x > 0$ and $y > 0$), and the other lies in the third quadrant (where $x < 0$ and $y < 0$). Both branches are smooth curves that get closer and closer to the $x$-axis and the $y$-axis but never touch or cross them — those axes are the asymptotes of the graph.

The physical idea the figure teaches is that $f(x) = 1/x$ is a reciprocal relationship: as $x$ grows large and positive, $y$ becomes very small and positive (the branch hugs the $x$-axis far to the right). As $x$ approaches $0$ from the positive side, $y$ shoots upward without bound (the branch hugs the $y$-axis near the top). The same behaviour repeats in the third quadrant but with both coordinates negative: as $x$ becomes large and negative, $y$ is a small negative number; as $x$ approaches $0$ from the negative side, $y$ plunges downward without bound. The graph is symmetric about the origin — if you rotate it $180^\circ$, it lands on itself — which reflects the algebraic fact that $f(-x) = -f(x)$.

The key formula the textbook develops with this figure is the definition of the function itself:

Here, $\mathbb{R}$ denotes the set of all real numbers, and $\mathbb{R} - \{0\}$ means all real numbers except $0$. The domain of $f$ is $\mathbb{R} - \{0\}$ because division by zero is undefined. The range is also $\mathbb{R} - \{0\}$: no matter what non-zero real number you pick, there is some $x$ that gives that $y$ (for example, $y = 2$ comes from $x = 1/2$), but $y = 0$ is never achieved because $1/x$ can never equal zero.

The modulus function is defined piecewise because absolute value behaves differently for non‑negative and negative numbers. The definition is:

For every $x \ge 0$, the output is simply $x$ itself — so on the right half of the graph, the function follows the line $y = x$. For every $x < 0$, the output is the positive opposite $-x$, which means the left half follows the line $y = -x$. Both are straight lines through the origin, one with slope $+1$ and the other with slope $-1$.

The figure shows a standard Cartesian plane with the $x$-axis horizontal and the $y$-axis vertical. The graph is a single continuous V‑shaped curve. Its vertex sits exactly at the origin $(0,0)$. From the vertex, the left arm rises into the second quadrant (where $x$ is negative, $y$ is positive) at a $45^\circ$ angle to the axes — that is the line $y = -x$. The right arm rises into the first quadrant (both $x$ and $y$ positive) also at $45^\circ$ — that is the line $y = x$. The two arms are symmetric about the $y$-axis.

The physical idea the graph teaches is that absolute value measures distance from zero on the number line. Distance is always non‑negative, and the graph’s V‑shape reflects that: as you move away from zero in either direction, the distance (the $y$-value) increases at the same rate. The sharp corner at the origin is the key visual clue — the function changes its rule exactly at $x=0$, and that point is the minimum value of the function.

The signum function is a simple but important piecewise-defined function that tells you the sign of a real number. Its formal definition is:

The domain is all real numbers $\mathbb{R}$, and the range is the three-point set $\{-1, 0, 1\}$.

What the graph shows. The figure plots $y = \text{sgn}(x)$ on a standard Cartesian plane. For every positive $x$, the output is exactly $1$ — so the graph is a horizontal ray along $y = 1$ that starts just to the right of the origin and extends infinitely to the right. The starting point at $x = 0$ is an open circle, because the function does not take the value $1$ at $x = 0$; it takes $0$ there. An arrowhead on the right end of the ray indicates it continues without bound.

For every negative $x$, the output is $-1$, giving a horizontal ray along $y = -1$ that runs from the far left toward the origin. Again, the endpoint at $x = 0$ is an open circle — the value $-1$ is not assigned at $x = 0$. An arrowhead on the left end shows the ray extends to $-\infty$.

At the origin itself, the graph has a single filled dot at $(0, 0)$. This solid dot represents the third piece of the definition: $f(0) = 0$. The open circles on the two rays and the filled dot at the origin together make the graph complete and unambiguous — they show exactly which $y$-value belongs to each $x$.

The key idea the figure teaches. The signum function is a piecewise constant function that compresses the entire real line into just three outputs. It is often used in later mathematics to express the sign of a quantity without caring about its magnitude. For example, the absolute value function $|x|$ can be written as $|x| = x \cdot \text{sgn}(x)$, and the derivative of $|x|$ (for $x \neq 0$) is $\text{sgn}(x)$.

This alternative form is useful: for $x > 0$, $\frac{|x|}{x} = \frac{x}{x} = 1$; for $x < 0$, $\frac{|x|}{x} = \frac{-x}{x} = -1$. It fails at $x = 0$ because division by zero is undefined, which is why the piecewise definition is the complete one. …

The figure shows the graph of the function $f(x) = [x]$, where $[x]$ denotes the greatest integer less than or equal to $x$. The horizontal axis is labelled $x$ and runs from $-3$ to $5$; the vertical axis is labelled $y$ and runs from $-3$ to $3$. The graph consists of seven horizontal step segments, each exactly one unit wide. Every segment is drawn in indigo.

For each integer $n$, the segment corresponding to $[x] = n$ lies at height $y = n$ and covers the interval $n \leq x < n+1$. The left endpoint of each segment — the point $(n, n)$ — is marked with a filled dot, indicating that the value $n$ is included at that $x$. The right endpoint — the point $(n+1, n)$ — is marked with an open circle, indicating that the value $n$ is not attained at $x = n+1$ (because at $x = n+1$, the greatest integer less than or equal to $x$ jumps to $n+1$). The graph therefore rises by one unit at every integer value of $x$, creating a staircase that climbs to the right and descends to the left.

The key idea the figure teaches is that the greatest integer function is a step function — it is constant on intervals between consecutive integers, and it jumps discontinuously at every integer. The filled and open dots make the left‑continuity of the function visible: at each integer $x = n$, the function value is $n$ (the closed dot), but just to the right of $n$ the value is still $n$ until the next integer is reached.