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If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Please disable adblock in order to continue browsing our website.Unfortunately, in the last year, adblock has now begun disabling almost all images from loading on our site, which has lead to mathwarehouse becoming unusable for adlbock users. Here we will learn about function notation, including different forms of function notation, how to evaluate functions for given values and how to manipulate algebraic expressions using functions. There are also function notationworksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. What is function notation?Function notation is a way of expressing a relationship between two variables. We are used to writing equations of straight lines in the form y = mx + c . The x is the
input value known as the independent variable. To use function notation we just substitute the values of x into the expression and evaluate it. There are different types of function notation. As well as examples like g(x) = x^2 + 2 , we may also see it given as g : x → x^2 + 2 . This type of function notation that is more common in A level mathematics. Function notation is also used in the table function of a scientific calculator. The table function is useful for finding values when graphing linear equations, quadratics, cubics and other polynomials. What is function notation?How to use function notationIn order to evaluate a function using function notation:
How to use function notation.Function notation worksheetGet your free function notation worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Function notation worksheetGet your free function notation worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Function notation examplesExample 1: evaluating linear functions for numerical valuesFind f(3) when f(x)= 4x - 1
f\left( {} \right)=4\left( {} \right)-1 2Replace the x in the function with the number or algebraic term in the brackets next to the name of the function. f\left( 3 \right)=4\left( 3 \right)-1 3Apply the correct operations to the number or term as appropriate and simplify. f\left( 3 \right)=11 Example 2: evaluating quadratic functions for numerical valuesFind g(-2) when g(x) = x^2 + 4 Write out the function for x using function notation, replacing the x with an empty set of brackets. g\left( {} \right)={{\left( {} \right)}^{2}}+4 Replace the x in the function with the number or algebraic term in the brackets next to the name of the function. g\left( -2 \right)={{\left( -2 \right)}^{2}}+4 Apply the correct operations to the number or term as appropriate and simplify. g\left( -2 \right)=8 Example 3: evaluating cubic functions for numerical valuesFind h(8) when h(x) = x^3 - 3x Write out the function for x using function notation, replacing the x with an empty set of brackets. h\left( {} \right)={{\left( {} \right)}^{3}}-3\left( {} \right) Replace the x in the function with the number or algebraic term in the brackets next to the name of the function. h\left( 8 \right)={{\left( 8 \right)}^{3}}-3\left( 8 \right) Apply the correct operations to the number or term as appropriate and simplify. h\left( 8 \right)=488 Example 4: evaluating linear functions for algebraic expressionsFind f(2m) when f(x) = 5x + 7 Write out the function for x using function notation, replacing the x with an empty set of brackets. f\left( {} \right)=5\left( {} \right)+7 Replace the x in the function with the number or algebraic term in the brackets next to the name of the function. f\left( 2m \right)=5\left( 2m \right)+7 Apply the correct operations to the number or term as appropriate and simplify. f\left( 2m \right)=10m+7 Example 5: evaluating quadratic functions for algebraic expressionsFind g(a+3) when g(x) = x^2 - 1 Write out the function for x using function notation, replacing the x with an empty set of brackets. g\left( {} \right)={{\left( {} \right)}^{2}}-1 Replace the x in the function with the number or algebraic term in the brackets next to the name of the function. g\left( a+3 \right)={{\left( a+3 \right)}^{2}}-1 Apply the correct operations to the number or term as appropriate and simplify. \begin{aligned} & g\left( a+3 \right)={{a}^{2}}+6a+9-1 \\\\ & g\left( a+3 \right)={{a}^{2}}+6a+8 \end{aligned} Common misconceptions
It is common for f(x) to be thought of as
“ f times x ” rather than “ f of x ”. E.g. Whereas the correct solution is: \begin{aligned} &f(x)=x+3 \\ &f(2)=2+3 \\ &f(2)=5 \end{aligned} Practice function notation questionsReplace the x with 4 and simplify.
Replace the x with 7 and simplify. Replace the x with -5 and simplify.
Replace the x with 3n , expand the brackets and simplify. Replace the x with 4k+1 , expand the brackets and simplify. Replace the x with a-2 , expand the brackets and simplify. Function notation GCSE questions1. Given that f(x)=5x-2 (a) Find f(-4) (b) Find x when f(x)=8 (3 marks) Show answer (a) -22 (1) (b) Equation formed 5x-2=8 (1) Answer x=2 (1) 2. (a) If f(x)=4x+1 , write a simplified expression for f(2a-1) (b) If h(x)=x^2+2x-3 , write a simplified expression for h(2m+3) (5 marks) Show answer (a) Substitution seen 4(2a-1)+1 (1) 8a-3 (1) (b) Substitution seen (2m+3)^2+2(2m+3)-3 (1) Expanded brackets 4m^2+12m+9+4m+6-3 (1) Simplified expression 4m^2+16m+12 (1) 3. Given that f(x)=x^2+2 and g(x)=3(x+4). Find the value of x which satisfies f(x)=g(x) (4 Marks) Show answer Set equal x^2+2=3(x+4) (1) Form quadratic x^2-3x-10=0 (1) Factorise (x-5)(x+2)=0 (1) Both solutions x=5,-2 (1) Learning checklistYou have now learned how to:
Still stuck?Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. Find out more about our GCSE maths revision programme. What is the evaluation of functions?Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4.
What is function notation?Function notation is a way to write functions that is easy to read and understand. Functions have dependent and independent variables, and when we use function notation the independent variable is commonly x, and the dependent variable is F(x).
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