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
\[f(x)=\text{the output produced by input }x\]
- Read the brackets as an input The value inside the brackets is the input. The function notation denotes the corresponding output \(\displaystyle f(x)=2x+1\Rightarrow f(4)=2(4)+1=9\)
- Substitute every occurrence Replace every occurrence of the input variable before simplifying \(\displaystyle f(x)=x^2+3x+1\Rightarrow f(2)=2^2+3(2)+1=11\)
- Solve for a preimage When an output is given, set the rule equal to that output and solve for the input \(\displaystyle f(x)=2x+4,\quad f(x)=10\Rightarrow2x+4=10\Rightarrow x=3\)
- Use independent conditions to fit a rule Use enough independent equations to determine every unknown coefficient, then check each supplied condition \(\displaystyle g(x)=ax+b,\quad g(2)=7,\quad g(5)=16\Rightarrow a=3,\ b=1\)
Worked example
The linear rule has two unknown coefficients and satisfies both displayed conditions. Determine the rule, then evaluate it at input three
\[g(x)=ax+b,\quad g(2)=7,\quad g(5)=16\]
- \(\displaystyle 2a+b=7,\quad 5a+b=16\)Substitute each supplied input into the linear rule
- \(\displaystyle 3a=9\Rightarrow a=3\)Subtract the first equation from the second to eliminate the constant term
- \(\displaystyle 2(3)+b=7\Rightarrow b=1,\quad g(x)=3x+1\)Substitute the recovered slope into either condition to find the constant term and write the rule
- \(\displaystyle g(3)=3(3)+1=10\)Substitute the requested input into the recovered rule
Common mistakes
Substituting one occurrence
✕\(\displaystyle f(x)=x^2+3x,\quad f(2)=2^2+3x=4+3x\)✓\(\displaystyle f(x)=x^2+3x,\quad f(2)=2^2+3(2)=10\)
Replace every occurrence of the input variable before simplifyingFunction notation is read as a product
✕\(\displaystyle f(x)=2x+1,\quad f(4)=4f\)✓\(\displaystyle f(x)=2x+1,\quad f(4)=2(4)+1=9\)
The function brackets supply an input; they do not denote multiplicationA 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 uniquelyDifficulty 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
How do function brackets differ from multiplication?+
Function brackets specify the input used to produce an output. Multiplication combines numerical factors; a function name is not a numerical factor
What is a preimage?+
A preimage is an input that produces a stated output. Set the function rule equal to the required output and solve for the input
How many conditions determine a polynomial rule?+
There is one coefficient for each power, including the constant term. For example, a quadratic has three coefficients and needs three independent conditions. Check that the recovered leading coefficient is nonzero when the degree is specified exactly
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 12: Continuous probability distributions
In this chapter
12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables