IB Applications and Interpretation SL · Functions · Section 1 of 7

Functions: domain, range and features of the graph

3 worked examples · By Álvaro López

Key idea

A function assigns to each input value $x$ a single output value $f (x)$ . To describe it in the exam you must be able to say for which values of $x$ it exists (the domain), which values it takes (the range) and what its graph does: where it cuts the axes, where it rises or falls and where it has maxima and minima.

Function, domain and range

A function $f$ is a rule that assigns to each number $x$ in a set exactly one number $y = f (x)$ . The variable $x$ is the independent variable (the one you choose) and $y$ is the dependent variable (the one that results). If $f (x) = 3 x - 2$ , then $f (4) = 10$ : the number $10$ is the image of $4$ .

The domain depends on the formula and also on the context. In $f (x) = \frac{10}{x - 4}$ you cannot divide by zero, so $x \neq = 4$ . In a cost model, $x$ may be the number of units sold and only makes sense with $x \geq 0$ . A question may also restrict it explicitly, as in $0 \leq x \leq 8$ .

Intervals

To write domains and ranges, intervals are used. An included endpoint is written with a square bracket facing into the interval, and an excluded one with a square bracket facing out of it; infinity is never included.

Inequality	Interval	Meaning
$2 \leq x \leq 5$	$[2, 5]$	includes 2 and 5
$2 < x \leq 5$	$] 2, 5]$	excludes 2, includes 5
$x \geq 2$	$[2, \infty [$	from 2 onwards
$x < 5$	$] - \infty, 5 [$	all values less than 5

To join two intervals, use $\cup$ . For example, $x \neq = 4$ is written $] - \infty, 4 [\cup] 4, \infty [$ . The set of all real numbers is $R$ .

Graph of a function

The graph of $f$ is the set of points $(x, f (x))$ on a set of axes. To sketch it by hand, make a table of values, plot the points and join them with a line. In the exam you will normally obtain it with the GDC.

To read the domain from a graph, look at which values of $x$ the curve covers from left to right. To read the range, look at which heights $y$ it reaches from bottom to top. If the endpoint is a filled dot, it is included; if it is an open dot, it is not.

Intercepts with the axes and sign

The $y$ -intercept is the point $(0, f (0))$ . There is at most one, because $x = 0$ gives a single value.

The $x$ -intercepts are the points where $f (x) = 0$ . Their $x$ -coordinates are called the zeros of the function, or the roots of the equation $f (x) = 0$ . There may be none, one or several.

The sign of a function tells where $f (x) > 0$ (the graph is above the $x$ -axis) and where $f (x) < 0$ (below it). The zeros separate those regions.

Increase, maxima and minima

A function is increasing on an interval if, as you move to the right, the graph rises; it is decreasing if it falls. These intervals are always given in terms of values of $x$ .

A local maximum is a point higher than all the points around it; a local minimum is one lower than all the points around it. Each is given with both its coordinates. The greatest value of $y$ on the whole graph is the absolute maximum (if it exists), and the least is the absolute minimum.

Graph of f(x) = x³ − 3x with a local maximum at (−1; 2), a local minimum at (1; −2) and x-intercepts at −√3, 0 and √3; arrows below the axis showing increasing, decreasing and increasing

Symmetry: even and odd functions

A function is even if $f (- x) = f (x)$ for all $x$ in the domain: its graph is symmetric about the $y$ -axis (like $x^{2}$ or $x^{4}$ ). It is odd if $f (- x) = - f (x)$ : its graph is symmetric about the origin (like $x$ or $x^{3}$ ). To check, substitute $- x$ for $x$ and simplify. Many functions are neither even nor odd.

Asymptotes: a first idea

An asymptote is a line that the graph approaches more and more closely without ever touching it. The function $f (x) = \frac{10}{x - 4}$ has a vertical asymptote at $x = 4$ , because as $x$ approaches $4$ the value $∣ f (x) ∣$ grows without limit, and a horizontal asymptote at $y = 0$ , because for very large $x$ , $f (x)$ approaches $0$ . Asymptotes limit the domain or the range: here the range does not include $0$ . You will see them in detail in the section on integer powers.

Worked examples

Example 1 Basic

Value, domain and range

A function is defined as $f (x) = 5 - 2 x$ for $- 1 \leq x < 3$ .

a) Calculate $f (2)$ .

b) Write the range of $f$ as an interval.

Solution

a) Substitute: $f (2) = 5 - 2 \cdot 2 = 1$ .

b) The function is a decreasing line, so the greatest value is at the left end of the domain and the least at the right end: $f (- 1) = 5 + 2 = 7 f (3) = 5 - 6 = - 1$ As $x = - 1$ is included, the $7$ is reached. As $x = 3$ is not included, the $- 1$ is not reached. The range is $] - 1, 7]$ .

Example 2 Medium

Reading the features of a bounded parabola

Let $g (x) = x^{2} - 4 x$ , with domain $- 1 \leq x \leq 5$ .

a) Find the $x$ -intercepts.

b) Find the coordinates of the minimum of $g$ (you may use the GDC).

c) Write the range of $g$ .

d) Write the interval of $x$ on which $g$ is increasing.

Solution

a) Solve $g (x) = 0$ : $x (x - 4) = 0$ , so $x = 0$ or $x = 4$ . Both values are inside the domain. The intercepts are $(0, 0)$ and $(4, 0)$ .

b) Draw the graph with the GDC and find the lowest point: it is $(2, - 4)$ . Check by hand, because $g (2) = 4 - 8 = - 4$ .

c) The minimum value is $- 4$ . The maximum may be at an end of the domain, so we calculate it: $g (- 1) = 1 + 4 = 5 g (5) = 25 - 20 = 5$ The range is $[- 4, 5]$ .

d) The graph falls from $x = - 1$ to $x = 2$ and rises from $x = 2$ to $x = 5$ . It is increasing on $[2, 5]$ .

Piece of the parabola g(x) = x² − 4x between x = −1 and x = 5, with endpoints (−1; 5) and (5; 5), minimum (2; −4), x-intercepts at 0 and 4, axis of symmetry x = 2 and a shaded band for the range between −4 and 5

Example 3 Hard

Symmetry, zeros and sign of a quartic function

Let $h (x) = x^{4} - 5 x^{2} + 4$ .

a) Calculate $h (3)$ and $h (- 3)$ , and state whether the function is even, odd or neither.

b) Find the zeros of $h$ .

c) Find the values of $x$ for which $h (x) < 0$ .

d) Using the GDC, find the minimum value of $h$ .

Solution

a) $h (3) = 81 - 45 + 4 = 40$ and $h (- 3) = 81 - 45 + 4 = 40$ . In general, as powers with even exponent do not change when the sign of $x$ changes, $h (- x) = x^{4} - 5 x^{2} + 4 = h (x)$ for all $x$ . The function is even and its graph is symmetric about the $y$ -axis.

b) Let $u = x^{2}$ : $u^{2} - 5 u + 4 = 0$ , which factorises as $(u - 1) (u - 4) = 0$ , so $u = 1$ or $u = 4$ . Then $x^{2} = 1$ or $x^{2} = 4$ , and the zeros are $x = - 2, - 1, 1, 2$ . The GDC gives the same four values.

c) Factorise: $h (x) = (x - 2) (x - 1) (x + 1) (x + 2)$ . Between two consecutive zeros the sign does not change, and it changes at each zero. For $x > 2$ all four factors are positive, so $h > 0$ . On crossing $x = 2$ it becomes negative, on crossing $x = 1$ it becomes positive again, on crossing $x = - 1$ it becomes negative again and on crossing $x = - 2$ , positive. Therefore $h (x) < 0 ⟺ x \in] - 2, - 1 [\cup] 1, 2 [$

d) Draw $h$ on the GDC and look for the lowest point. The graph has two equal minima, symmetric about the $y$ -axis, at $x = \pm \frac{10}{2} \approx \pm 1.58$ . At both, $h = - \frac{9}{4} = - 2.25$ . The minimum value of $h$ is $- 2.25$ .

In the exam: an answer about sign is marked for the correct intervals, not for the zeros. Write the intervals with the endpoints excluded, because at the zeros $h = 0$ and it is not negative. If asked for "the minimum", give the $y$ -value and, if they ask for the point, also the $x$ -value.

Common mistakes

Summary

A function gives a single $y$ for each $x$ ; the vertical line test checks this on a graph.
The domain is the possible values of $x$ (from the formula, the context or the question); the range is the values of $y$ that are reached.
Intervals use outward-facing brackets for endpoints that are not included, and $\cup$ to join intervals.
$y$ -intercept: $(0, f (0))$ . Zeros: solutions of $f (x) = 0$ .
Increase is given in intervals of $x$ ; maxima and minima as points with two coordinates.
$f$ is even if $f (- x) = f (x)$ (symmetry about the $y$ -axis) and odd if $f (- x) = - f (x)$ (symmetry about the origin).
With the GDC you draw, find intercepts, maxima and minima; you interpret the result.

Keep studying

Maths classes in Aldaia and online

Want to work on it together?

Free first consultation: we look at your level and your exam, and build a plan. Álvaro replies on WhatsApp.

Message us on WhatsApp