1. Functions

Learning objectives

1.1Understand function terminology• Understand the terms: function, domain, range (image set), one-one function, many-one function, inverse function and composition of functions.• Includes explaining in words why a given relation is a function.1.2Find domains and ranges of functions• Find the domain and range of functions, including inverse and composite functions.• Restrict the domain of f where needed for f^-1 and/or gf to exist.• Understand that the domain of gf is a subset of the domain of f, and the range of gf is a subset of the range of g.1.3Use function notation• Recognise and use function notation such as f(x), f: x -> lg x, f^-1(x), fg(x) = f(g(x)) and f^2(x) = f(f(x)).• The notation f^2(x) will not be used with trigonometric functions.1.4Relate y = f(x) and y = |f(x)|• Understand the relationship between y = f(x) and y = |f(x)| where f(x) may be linear, quadratic, cubic or trigonometric.• For trigonometric functions, use y = a sin bx + c, y = a cos bx + c or y = a tan bx + c, where a is a positive integer, b is a simple fraction or integer, and c is an integer.• Fractions for b have denominator 2, 3, 4, 6 or 8 only.1.5Explain when inverses do not exist• Explain in words why a given function does not have an inverse.1.6Find inverse functions• Find the inverse of a one-one function.• Use correct inverse-function notation.1.7Form and use composite functions• Form and use composite functions.• Understand that the order of functions is important, so fg may not be the same as gf.1.8Sketch functions and inverses• Use sketch graphs to show the relationship between a function and its inverse.• Understand that each graph is the reflection of the other in the line y = x.

A function assigns exactly one output to every input in its domain. Different inputs may share an output, but one input cannot have two outputs.

Term	Meaning
domain	permitted input values
range (image set)	outputs actually produced
one-one	distinct inputs give distinct outputs
many-one	two or more inputs may give the same output
inverse	reverses a one-one function
composition	uses one function's output as another's input

A graph represents a function when every vertical line meets it at most once. It is one-one when every horizontal line also meets it at most once; a many-one function still passes the vertical-line test but fails the horizontal-line test.

‘Exactly one output’ is required only for inputs in the stated domain. Having several inputs map to one output does not stop a relation being a function; it stops that function being one-one.

A domain records allowed inputs and a range records the resulting outputs. Both must respect algebraic restrictions and any stated interval.

Expression feature	Domain condition
denominator $q(x)$	$q(x)\ne0$
square root $\sqrt{q(x)}$	$q(x)\geq0$
$\ln q(x)$ or $\lg q(x)$	$q(x)>0$

For an inverse, domain and range swap: $\operatorname{Dom}(f^{-1})=\operatorname{Range}(f)$ and $\operatorname{Range}(f^{-1})=\operatorname{Dom}(f)$ . Restrict a many-one formula to an interval on which it is one-one before defining its inverse.

For $gf(x)=g(f(x))$ , first require $x$ to be in the domain of $f$ , then require $f(x)$ to lie in the domain of $g$ . Therefore $\operatorname{Dom}(gf)\subseteq\operatorname{Dom}(f)$ and $\operatorname{Range}(gf)\subseteq\operatorname{Range}(g)$ .

If $f(x)=x^2$ is restricted to $x\geq0$ , then its range is $[0,\infty)$ and $f^{-1}(x)=\sqrt{x}$ has domain $x\geq0$ . Without the restriction, $x^2$ is many-one and has no inverse function on all real numbers.

State inequalities with the correct strictness. A forbidden denominator value is excluded, while a zero radicand is allowed for a square root.

Notation	Meaning
$f(x)$	output of $f$ at input $x$
$f:x\mapsto\lg x$	$f$ maps $x$ to $\lg x$
$f^{-1}(x)$	inverse-function output
$fg(x)$	$f(g(x))$ : apply $g$ first
$f^2(x)$	$f(f(x))$ : apply $f$ twice

If $f(x)=x^2+1$ and $g(x)=3x-2$ , then $fg(4)=f(g(4))=f(10)=101$ . For iteration, $f^2(2)=f(f(2))=f(5)=26$ .

$f^{-1}(x)$ is not $1/f(x)$ , and $f^2(x)$ is not generally $[f(x)]^2$ . Read adjacent function letters from right to left. In this syllabus, $f^2(x)$ notation is not used with trigonometric functions.

The graph of $y=|f(x)|$ keeps every part of $y=f(x)$ on or above the $x$ -axis and reflects every part below it across the $x$ -axis.

$(x,y)\mapsto(x,|y|)$

The domain and $x$ -intercepts stay unchanged, all output values become non-negative, and a crossing of the $x$ -axis usually becomes a sharp turning point. Apply the rule point by point to permitted linear, quadratic, cubic or trigonometric $f(x)$ .

For $f(x)=5x-7$ , the zero is $x=7/5$ . Thus $y=|5x-7|$ is V-shaped with vertex $(7/5,0)$ and $y$ -intercept $(0,7)$ ; the negative half of the original line is reflected upward.

Absolute value changes output signs, not input values: $|f(x)|$ is generally different from $f(|x|)$ . Do not reflect parts that are already above the axis.

An inverse function exists only when the original function is one-one on its stated domain.

Reversing a many-one function makes one former output point to several former inputs. That reversed relation fails the function rule because one input would have more than one output. Graphically, the original fails the horizontal-line test, so its reflection in $y=x$ fails the vertical-line test.

A suitable domain restriction can select one one-one branch. For example, $f(x)=x^2$ has no inverse on $\mathbb R$ , but on $x\geq0$ it has inverse $f^{-1}(x)=\sqrt{x}$ .

Do not say only ‘because it is quadratic’. The reason is that distinct permitted inputs give the same output; identify the many-one behaviour or a failed horizontal-line test.

To invert a one-one function: write $y=f(x)$ , rearrange to make $x$ the subject, then interchange the variable names and state the inverse domain. Any branch choice must match the original domain restriction.

$f(x)=(x-1)^2+3,\ x\geq1\quad\Longrightarrow\quad f^{-1}(x)=1+\sqrt{x-3},\ x\geq3$

The positive square-root branch is required because the original restriction gives $x-1\geq0$ . The inverse domain $x\geq3$ is the original range, while the inverse range $x\geq1$ is the original domain.

Check both compositions on their valid domains: $f^{-1}(f(x))=x$ and $f(f^{-1}(x))=x$ . Use $f^{-1}$ notation only for the inverse function, never for a reciprocal.

In $fg(x)=f(g(x))$ , the right-hand function acts first. Composition is substitution: replace every $x$ in the outer function by the complete inner expression.

$f(x)=x^2+1,\ g(x)=3x+2\quad\Rightarrow\quad fg(x)=(3x+2)^2+1,\quad gf(x)=3(x^2+1)+2$

For a value, follow the same inside-first order: $fg(1)=f(g(1))=f(5)=26$ . For an equation such as $fg(x)=5$ , first form the correct composite, then solve while enforcing the composite domain.

The composite exists only when the input is valid for the inner function and its output is valid for the outer function. A logarithm, square root or denominator can therefore remove inputs after substitution.

Usually $fg\ne gf$ ; switching the letters changes both the expression and possibly the domain. Keep brackets around the entire inner expression during substitution.

The graphs of a one-one function and its inverse are reflections of each other in the line $y=x$ .

$(a,b)\text{ on }y=f(x)\quad\Longleftrightarrow\quad(b,a)\text{ on }y=f^{-1}(x)$

Swap the coordinates of useful points, then reflect the whole shape: an $x$ -intercept $(a,0)$ becomes the inverse's $y$ -intercept $(0,a)$ ; horizontal and vertical asymptotes swap; domain and range swap. Points on $y=x$ remain fixed and satisfy $f(x)=x$ .

Sketch $y=x$ as a guide, mark reflected intercepts, endpoints and asymptotes, and preserve the restricted domains. Label both curves so their roles are unambiguous.

A many-one graph cannot be reflected into an inverse function until its domain is restricted to a one-one branch. Reflection does not mean changing every coordinate's sign.

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—2. Quadratic functionsRead

1. Functions

Learning objectives

Understand functions and their key terms

Find domains and ranges, including restrictions

Read and use function notation

Transform y=f(x) into y=|f(x)|

Explain when an inverse function does not exist

Find an inverse function with the correct domain

Form and use composite functions in the right order

Sketch a function and its inverse

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