IB Analysis and Approaches HL · Functions · Section 2 of 15

Graphs and elementary functions

3 worked examples · By Álvaro López

Key idea

The graph of a function is the drawing of all its pairs $(x, f (x))$ . Being able to read it (domain, range, intercepts, maxima) and to draw the simplest functions, including piecewise ones, is the basis of almost every function problem in the IB exam.

The graph of a function

A point $(a, b)$ lies on the graph if and only if $f (a) = b$ . This is useful for two things: checking whether a point belongs to the curve (by substituting) and, conversely, deducing information from the formula using a point on the drawing.

Vertical line test. A curve is the graph of a function if no vertical line cuts it at more than one point. If a vertical line cut it twice, that input $x$ would have two outputs, and that is not a function. A circle, for example, does not pass the test.

Constant and linear functions

The constant function $f (x) = k$ always gives the same value. Its graph is a horizontal line and its range is just ${k}$ .

The linear function $f (x) = m x + c$ has a straight line as its graph. The number $m$ is the gradient (how much $y$ rises when $x$ increases by one unit), and $c$ is the y-intercept, the value of $y$ where the line cuts the $y$ -axis. If $m > 0$ the line rises, if $m < 0$ it falls, and if $m = 0$ it is constant.

A non-constant linear function has domain and range both equal to $R$ .

Piecewise functions

Some functions use a different formula depending on the value of $x$ . They are written with a brace:

$f (x) = ⎩ ⎨ ⎧ x + 3 3 - 2 x + 7 if - 3 \leq x < 0 if 0 \leq x \leq 2 if 2 < x \leq 5$

To evaluate it, first find which piece $x$ falls in, and use only the formula for that piece. To draw it, make a table with the endpoints of each piece and join each piece separately.

Graph of a piecewise function made of three joined segments: from (-3,0) to (0,3), horizontal up to (2,3), and from (2,3) down to (5,-3)

Reading a graph

From a graph you can read off several things directly:

Domain: the values of $x$ that the drawing covers, from left to right.
Range: the values of $y$ that it covers, from bottom to top.
Intercepts with the axes: with the $y$ -axis at $(0, f (0))$ ; with the $x$ -axis at the zeros, where $f (x) = 0$ .
Maximum and minimum: the highest and lowest values of $y$ reached (the height, not the position).
Solutions of $f (x) = k$ : the $x$ -coordinates of the points where the horizontal line $y = k$ cuts the curve.

Worked examples

Example 1 Basic

A line through two points

The graph of a linear function $f$ passes through $(1, 5)$ and $(4, - 1)$ .

a) Find the gradient.

b) Find the formula for $f (x)$ .

c) Find the point where the line cuts the $x$ -axis.

Solution

a) $m = \frac{- 1 - 5}{4 - 1} = \frac{- 6}{3} = - 2$ .

b) Use $y - 5 = - 2 (x - 1)$ , that is, $y = - 2 x + 7$ . Check with the other point: $- 2 \cdot 4 + 7 = - 1$ .

c) Solve $- 2 x + 7 = 0$ , which gives $x = \frac{7}{2}$ . The intercept is $(\frac{7}{2}, 0)$ .

Example 2 Medium

Evaluating and reading a piecewise function

Let $f (x) = ⎩ ⎨ ⎧ x + 3 3 - 2 x + 7 if - 3 \leq x < 0 if 0 \leq x \leq 2 if 2 < x \leq 5$

a) Calculate $f (- 1)$ , $f (1)$ and $f (4)$ .

b) Find the domain of $f$ .

c) Find the range of $f$ .

d) Solve $f (x) = 1$ .

Solution

a) $x = - 1$ is in the first piece: $f (- 1) = - 1 + 3 = 2$ . $x = 1$ is in the second: $f (1) = 3$ . $x = 4$ is in the third: $f (4) = - 8 + 7 = - 1$ .

b) The pieces cover from $- 3$ to $5$ with no gaps: domain $[- 3, 5]$ .

c) The first piece gives values from $0$ (included, at $x = - 3$ ) up to $3$ (not reached, at $x = 0$ ). The second gives only the value $3$ . The third falls from $3$ (not reached) down to $- 3$ (at $x = 5$ , included). Putting everything together, the range is $[- 3, 3]$ .

d) Solve on each piece:

first piece: $x + 3 = 1 \Rightarrow x = - 2$ , which lies in $[- 3, 0 [$ , so it is valid;
second piece: $3 = 1$ is impossible;
third piece: $- 2 x + 7 = 1 \Rightarrow x = 3$ , which lies in $] 2, 5]$ , so it is valid.

The solutions are $x = - 2$ and $x = 3$ .

Example 3 Hard

Mobile data tariff

A mobile operator charges the tariff $C (d)$ , in euros, according to the gigabytes $d$ consumed in a month: $C (d) = ⎩ ⎨ ⎧ 12 3 d - 3 27 if 0 \leq d \leq 5 if 5 < d \leq 10 if d > 10$

a) Calculate $C (3)$ , $C (8)$ and $C (15)$ .

b) Find the range of $C$ .

c) A customer paid $18$ euros this month. How many gigabytes did they use?

d) Find the values of $d$ for which the bill is at its maximum possible value.

Solution

a) $C (3) = 12$ (first piece), $C (8) = 3 \cdot 8 - 3 = 21$ (second piece) and $C (15) = 27$ (third piece).

b) On the second piece the charge rises from $3 \cdot 5 - 3 = 12$ (not reached, at $d = 5$ ) up to $3 \cdot 10 - 3 = 27$ (at $d = 10$ , included). The first and third pieces give the values $12$ and $27$ . There are no jumps: at $d = 5$ the second piece gives $12$ , the same as the first, and at $d = 10$ it gives $27$ , the same as the third. The range is $[12, 27]$ .

c) $12 < 18 < 27$ , so the value lies in the second piece: $3 d - 3 = 18 \Rightarrow d = 7$ gigabytes.

d) The maximum bill is $27$ . It is obtained at $d = 10$ (second piece) and throughout the third piece ( $d > 10$ ): $d \geq 10$ , that is, $[10, + \infty [$ .

In the exam: when a piecewise function comes from a context, check the value at each join point (here $d = 5$ and $d = 10$ ) and say whether there is a jump or not. When finding the range, look at the value of each piece and put them together, without forgetting the constant pieces.

Common mistakes

Summary

The graph of $f$ is the set of points $(x, f (x))$ ; a curve is the graph of a function if it passes the vertical line test.
Linear function $m x + c$ : gradient $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , y-intercept $c$ .
A piecewise function uses the formula of the piece in which $x$ falls; the point is filled if the endpoint is included and hollow if not.
From a graph you can read the domain, range, intercepts with the axes, maximum and minimum.
At a join point, check whether the two pieces match (no jump) or not.

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