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Functions and their graphs

A function assigns each input value a single output value, and its graph shows it at a glance: where it cuts the axes, where it increases, its maxima and its asymptotes. This topic explains domain and range, transformations of a graph, asymptotes of rational functions, the inverse function and curves that are not functions (implicit, parametric and polar).

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Domain and range

The domain is the set of $x$ for which the formula makes sense: remove the zeros of denominators, whatever makes the radicand of an even root negative and whatever makes the argument of a logarithm $\leq 0$ . The range is the set of $y$ values the function reaches.

Function	Domain	Range
$x^{2}$	$R$	$[0, \infty)$
$x$	$[0, \infty)$	$[0, \infty)$
$1/ x$	$R ∖ {0}$	$R ∖ {0}$
$e^{x}$	$R$	$(0, \infty)$
$ln x$	$(0, \infty)$	$R$
$sin x$	$R$	$[- 1, 1]$

Transformations

Knowing the graph of $f$ , others are obtained without computing again:

Formula	Effect on the graph
$f (x) + k$	moves up $k$ units (down if $k < 0$ )
$f (x - h)$	moves $h$ to the right (the sign is misleading)
$a f (x)$	stretches vertically by $a$
$f (b x)$	compresses horizontally by $b$
$- f (x)$ , $f (- x)$	reflection in the $x$ axis, reflection in the $y$ axis

Asymptotes of rational functions

In $f = P / Q$ , each root of $Q$ that is not a root of $P$ gives a vertical asymptote: the function goes to $\pm \infty$ . At infinity the highest-degree terms dominate: if $P$ and $Q$ have the same degree, there is a horizontal asymptote at the ratio of their leading coefficients; if $P$ has lower degree, at $y = 0$ ; if it has one degree more, there is a slant asymptote (the division gives the line).

Inverse function

If $f$ is one-to-one (it does not repeat values), it has an inverse $f^{- 1}$ : it undoes what $f$ does, $f^{- 1} (f (x)) = x$ . It is found by solving $y = f (x)$ for $x$ and swapping the letters. Its graph is the reflection of that of $f$ in the line $y = x$ ; the domain of one is the range of the other.

Curves that are not functions

A circle is not the graph of a function: one $x$ corresponds to two $y$ . It is described as an implicit curve ( $x^{2} + y^{2} = 9$ ), in parametric form ( $x = 3 cos t$ , $y = 3 sin t$ ) or in polar form ( $r = 3$ ). The graphing calculator draws all three, as well as inequalities and piecewise functions.

When you tap a row, the graphing calculator looks for roots, maxima, minima, the

y

-intercept and the intersections with the other curves on

- 10 \leq x \leq 10

, and it only marks an extremum if it truly is one (not the «peaks» next to an asymptote).

Common mistakes

Moving $f (x + 2)$ to the right: it goes 2 to the left.
Confusing $f^{- 1} (x)$ with $1/ f (x)$ .
Giving as a vertical asymptote a root that also cancels the numerator: there is a hole there, not an asymptote.
Forgetting that a curve can cross its horizontal asymptote (what it does not do is cross a vertical one).

Worked examples

Example 1Shifting a graph

f (x) = x^{3} - 3 x

has a maximum at

(- 1, 2)

and a minimum at

(1, - 2)

. Where are those of

g (x) = f (x - 1) + 2

? Where do

f

and

g

meet?

Solution

$g$ is $f$ moved 1 to the right and 2 up: each point $(a, b)$ goes to $(a + 1, b + 2)$ . Maximum $(0, 4)$ , minimum $(2, 0)$ .
Expanding, $g (x) = x^{3} - 3 x^{2} + 4 = (x + 1) (x - 2)^{2}$ : roots $x = - 1$ and $x = 2$ (double: the graph touches the axis at the minimum).
Crossings: $x^{3} - 3 x = x^{3} - 3 x^{2} + 4$ → $3 x^{2} - 3 x - 4 = 0$ → $x = \frac{3 \pm 57}{6}$ , that is, $x = - 0.758306$ and $x = 1.75831$ .

g

: maximum

(0, 4)

, minimum

(2, 0)

; they meet at

(- 0.758306, 1.83887)

and

(1.75831, 0.16113)

Solve it in Simuveliq

In Math → Graphing calculator type f(x)=x^3-3x and g(x)=f(x-1)+2, one per row (with + Expression), and tap the row of $g$ . Below it appear «Roots: x = −1, x = 2», «Maxima: (0, 4)», «Minima: (2, 0)» and «Intersections: (−0.758306, 1.83887), (1.75831, 0.16113)».

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Example 2Rational function: domain, range and asymptotes

Study

f (x) = \frac{2 x + 1}{x - 1}

: domain, range, asymptotes and intercepts.

Solution

Domain: $x - 1 \neq = 0$ → $R ∖ {1}$ . At $x = 1$ the numerator is 3: vertical asymptote $x = 1$ .
Same degree on top and bottom: horizontal asymptote $y = 2/1 = 2$ .
Range: $y = \frac{2 x + 1}{x - 1}$ → $x = \frac{y + 1}{y - 2}$ , which exists except for $y = 2$ : range $R ∖ {2}$ (exactly the asymptote).
Intercepts: $f = 0$ when $2 x + 1 = 0$ → $x = - 0.5$ ; $f (0) = - 1$ .

Domain

R ∖ {1}

, range

R ∖ {2}

; asymptotes

x = 1

y = 2

; root

x = - 0.5

, intercept

(0, - 1)

Solve it in Simuveliq

In Math → Symbolic calculator, topic Calculus, operation Domain and range: type (2x+1)/(x-1) and press Compute. The Answer is $Dom f = (- \infty, 1) \cup (1, \infty)$ , $Range f = (- \infty, 2) \cup (2, \infty)$ .
In the Graphing calculator, y=(2x+1)/(x-1) gives «Roots: x = −0.5» and «y-intercept: (0, −1)».

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Example 3Line and circle

Where does the line

y = x + 1

cut the circle

x^{2} + y^{2} = 9

Solution

Substitute $y$ : $x^{2} + (x + 1)^{2} = 9$ → $2 x^{2} + 2 x - 8 = 0$ → $x^{2} + x - 4 = 0$ .
$x = \frac{- 1 \pm 17}{2}$ : $x = 1.56155$ ( $y = 2.56155$ ) and $x = - 2.56155$ ( $y = - 1.56155$ ).

(- 2.56155, - 1.56155)

(1.56155, 2.56155)

Solve it in Simuveliq

In Math → Graphing calculator type x^2+y^2=9 and y=x+1, one per row (with + Expression), and tap the row of the line. Below it appear «Intersections: (−2.56155, −1.56155), (1.56155, 2.56155)» (and its root, x = −1).

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Example 4The exponential and the logarithm

Check that

ln x

is the inverse of

e^{x}

and compare their intercepts.

Solution

$y = e^{x}$ → $x = ln y$ : swapping letters, $f^{- 1} (x) = ln x$ . Domain of $ln$ : $(0, \infty)$ , the range of $e^{x}$ .
$e^{x}$ cuts the $y$ axis at $(0, 1)$ and has no roots; $ln x$ has the root $x = 1$ , that is, the point $(1, 0)$ , and does not cut the $y$ axis. They are the same point with the coordinates swapped.
Neither meets $y = x$ : $e^{x} > x > ln x$ for all $x$ .

ln x = (e^{x})^{- 1}

;

(0, 1) \leftrightarrow (1, 0)

Solve it in Simuveliq

In Math → Graphing calculator type y=e^x, y=ln(x) and y=x, one per row (with + Expression), and tap the row of $ln x$ . Below it appear «Roots: x = 1» and no intersections; on the row of $e^{x}$ , «y-intercept: (0, 1)».

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Functions and their graphs

Domain and range

Transformations

Asymptotes of rational functions

Inverse function

Curves that are not functions

Common mistakes

Worked examples

Example 1Shifting a graph

Solution

Solve it in Simuveliq

Example 2Rational function: domain, range and asymptotes

Solution

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Example 3Line and circle

Solution

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Example 4The exponential and the logarithm

Solution

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