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Evaluate and determine function rules

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Substituting one occurrence. One of these lines is right. Which one?

Year 12, VCE Units 3 & 4, Chapter 7, 4 difficulty levels, Calculator allowedBuild a worksheet

Using function notation

Function notation names the output produced by a particular input. Substitute the input into every occurrence of the variable, or solve an equation when the required input is unknown

\[f(x)=\text{the output produced by input }x\]
  1. Read the brackets as an input The value in f(t) is the input t, and f(t) is the corresponding output \(\displaystyle f(x)=2x+1\Rightarrow f(4)=2(4)+1=9\)
  2. Substitute every occurrence Replace every x in the rule before simplifying \(\displaystyle f(x)=x^2+3x+1\Rightarrow f(2)=2^2+3(2)+1=11\)
  3. Solve for a preimage When an output is given, set f(x) equal to that output and solve for x \(\displaystyle f(x)=2x+4,\quad f(x)=10\Rightarrow2x+4=10\Rightarrow x=3\)
  4. Use independent conditions to fit a rule A rule with one unknown coefficient for each parameter needs one independent equation for each parameter, followed by a check \(\displaystyle g(x)=ax+b,\quad g(2)=7,\quad g(5)=16\Rightarrow a=3,\ b=1\)

Worked example

A linear function satisfies the conditions shown. Determine its rule and its output for input 3

\[g(x)=ax+b,\quad g(2)=7,\quad g(5)=16\]
  1. \(\displaystyle 2a+b=7,\quad 5a+b=16\)Substitute each supplied input into the linear rule
  2. \(\displaystyle 3a=9\Rightarrow a=3\)Subtract the first equation from the second to eliminate b
  3. \(\displaystyle 2(3)+b=7\Rightarrow b=1,\quad g(x)=3x+1\)Use a=3 in either condition, then write the rule
  4. \(\displaystyle g(3)=3(3)+1=10\)Substitute 3 into every occurrence of x in the recovered rule

Common mistakes

Substituting one occurrence
✕\(\displaystyle f(x)=x^2+3x,\quad f(2)=2^2+3x=4+3x\)✓\(\displaystyle f(x)=x^2+3x,\quad f(2)=2^2+3(2)=10\)
Every occurrence of x must be replaced before the expression is simplified
Function notation is read as a product
✕\(\displaystyle f(x)=2x+1,\quad f(4)=4f\)✓\(\displaystyle f(x)=2x+1,\quad f(4)=2(4)+1=9\)
f(4) is the output when the input is 4, not the product of f and 4
A rule is fitted from too few conditions
✕\(\displaystyle f(x)=ax^2+bx+c,\quad f(0)=1,\ f(1)=4\Rightarrow f(x)=3x+1\)✓\(\displaystyle f(x)=ax^2+bx+c\Rightarrow\text{three independent conditions are needed to determine }a,b,c\)
A quadratic rule has three unknown coefficients, so two conditions do not determine it uniquely

Difficulty levels

introEvaluate a linear or quadratic rule at integer inputs
standardFind a linear preimage or fit a linear rule from two image conditions
challengeRecover the coefficients of a quadratic or low-degree cubic from independent conditions
examWork through a mixed evaluation and preimage item with a formal mapping choice

Questions about this topic

What is the difference between f(3) and 3f?+

f(3) is the output of f for input 3. The expression 3f is a function whose output at each input is three times the output of f

What is a preimage?+

A preimage is an input that produces a stated output. Find it by solving an equation such as f(x)=10

How many conditions determine a polynomial rule?+

A polynomial has one more coefficient than its degree, so it needs the same number of independent input-output conditions to determine its rule

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 7: Antiderivatives

In this chapter

7A Antiderivatives and arbitrary constants: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Find complete families of polynomial anti-derivatives7B Power rule and integrand preparation: Recognise equivalent polynomial formsSimplify and combine algebraic fractionsAnti-differentiate a power of a linear function7C Integrating powers of linear functions: Recognise equivalent polynomial forms7C Integrating powers of linear functions: Solve a linear equationAnti-differentiate exponential and circular functionsAnti-differentiate by recognition from a supplied derivativeDetermine a polynomial from its derivative and a boundary conditionInfer primitive curves from a gradient functionReconstruct quantities from their ratesEvaluate and determine function rules7E Antiderivative families and applications: Solve a linear equationRound to decimal places and significant figuresScale a proportional quantity with the unitary method