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E2.13 Functions

Syllabus
0580–2028–2029
Topic
E2.13
Level
Extended

Learning objectives

E2.13.1Understand functions, domain and range and use function notation.• Understand functions, domain and range and use function notation.E2.13.2Understand and find inverse functions f –1(x).• Understand and find inverse functions f –1(x).E2.13.3Form composite functions as defined by gf(x) = g(f(x))• Form composite functions as defined by gf(x) = g(f(x)). Examples include: • f (x) = 3x – 5 • g(x) = 3(x + 4) • h(x) = 2x2 + 3. e.g. f(x) = 3 x + 2 and g(x) = (3x + 5)2. Find fg(x). Give your answer as a fraction in its simplest form. Candidates are not expected to find the domains and ranges of composite functions. This topic may include mapping diagrams.

Read functions, domains and ranges

A function assigns exactly one output to each allowed input. In f(x)f(x), xx is the input and f(x)f(x) is its output.

Term Meaning
domain all allowed input values
range all output values produced from that domain
f(a)f(a) substitute aa for every xx in the rule

f(x)=3x−5,domain={−3,0,2}f(x)=3x-5,\qquad \text{domain}=\{-3,0,2\}

Evaluate the rule for each domain value: f(−3)=−14f(-3)=-14, f(0)=−5f(0)=-5 and f(2)=1f(2)=1. Therefore the range is {−14,−5,1}\{-14,-5,1\}.

An input can be an expression. For example, f(x−3)=3(x−3)−5=3x−14f(x-3)=3(x-3)-5=3x-14. Substitute the whole expression in place of xx before simplifying.

To find an input from an output, form an equation. If f(x)=7f(x)=7, solve 3x−5=73x-5=7, giving x=4x=4. This reverses one evaluation; it is not yet a formula for the inverse function.

The range comes from the stated domain, not from every value the formula could accept. If two inputs give the same output, list that output only once in the range set.

Find and use an inverse function

An inverse function undoes the original function. If ff takes an input to an output, f−1f^{-1} takes that output back to the input.

Write y=f(x)y=f(x); interchange xx and yy; rearrange to make yy the subject; then write the result as f−1(x)f^{-1}(x).

f(x)=4x−1:x=4y−1⟹y=x+14⟹f−1(x)=x+14f(x)=4x-1:\quad x=4y-1\quad\Longrightarrow\quad y=\frac{x+1}{4}\quad\Longrightarrow\quad f^{-1}(x)=\frac{x+1}{4}

Check by composing the functions: f(f^{-1}(x))=4\left( rac{x+1}{4} ight)-1=x. Applying them in the opposite order also returns the starting value wherever both are defined.

If h−1(x)=5h^{-1}(x)=5, apply hh to both sides: x=h(5)x=h(5). This is often quicker than first finding an inverse formula.

The inverse reverses the mapping, so the original range becomes the inverse domain and the original domain becomes the inverse range.

f−1(x)f^{-1}(x) does not mean 1/f(x)1/f(x). The superscript −1-1 names the inverse operation; a reciprocal divides 11 by the function value.

Build composite functions in the correct order

A composite function uses the output of one function as the input of another. The function nearest xx is applied first.

gf(x)=g(f(x)),fg(x)=f(g(x))gf(x)=g(f(x)),\qquad fg(x)=f(g(x))

Expression First operation Second operation
gf(x)gf(x) find f(x)f(x) substitute into gg
fg(x)fg(x) find g(x)g(x) substitute into ff
ff(x)ff(x) find f(x)f(x) apply ff again

f(x)=2x+1,g(x)=x2+4:gf(x)=(2x+1)2+4=4x2+4x+5f(x)=2x+1,\quad g(x)=x^2+4:\qquad gf(x)=(2x+1)^2+4=4x^2+4x+5

For f(x)=3x+2f(x)=\dfrac{3}{x+2} and g(x)=(3x+5)2g(x)=(3x+5)^2, fg(x)=3(3x+5)2+2=13x2+10x+9fg(x)=\dfrac{3}{(3x+5)^2+2}=\dfrac{1}{3x^2+10x+9}. Substitute the entire inner expression, expand carefully, then simplify the fraction.

For a numerical composite such as fg(1)fg(1), calculate g(1)g(1) first and then use that result as the input to ff. Writing the intermediate value makes the order visible.

In general, gf(x)gf(x) and fg(x)fg(x) are different because the order changes. This syllabus does not require finding the domains or ranges of composite functions.