Here we will learn about inverse functions including what an inverse function is, the notation used for an inverse function and how to find an inverse function. Show
There are also inverse function worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. What are inverse functions?Inverse functions are functions which reverse or “undo” another function. To write the inverse of the function f , we use the notation f^{-1} . We have seen how to use a function machine to work backwards to find the input from a known output. If we were to write the algebraic expression relating to these inverse operations in the correct order, we would have found the inverse function. To find an inverse function we need to rewrite the function using y as the unknown variable and set the function equal to x. Then we need to rearrange the function to make y the subject and write the function using the inverse function notation. E.g. Inverse functions can be used to solve equations or find missing x values on graphs
if we know the y value. Inverse functions are also used when finding an unknown angle in a triangle using trigonometry. In A level mathematics instead of using sin^{-1},
cos^{-1} and tan^{-1}, the inverse trigonometric functions have different names. We would use arcsin, arccos and arctan. What are inverse functions?How to find inverse functionsIn order to find an inverse function:
Explain how to find inverse functionsInverse functions worksheetGet your free inverse functions worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Inverse functions worksheetGet your free inverse functions worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Inverse functions examplesExample 1: Finding the inverse of a linear functionIf f(x)=3x-7 , find f^{-1}(x).
3y-7=x 2Rearrange the equation to make y the subject. \begin{aligned} & 3y-7=x \\\\ & 3y=x+7 \hspace{2cm} \text{Add } 7 \text{ to both sides of the equation} \\\\ & y=\frac{x+7}{3} \hspace{2cm} \text{Divide both sides by } 3 \end{aligned} 3Write your inverse function using the f^{-1} notation. {{f}^{-1}}\left( x \right)=\frac{x+7}{3} Example 2: Finding the inverse of a linear functionIf g(x)=9-\frac{x}{2} find g^{-1}(x) . Write out the expression for the original function using a y instead of the x . Set this expression equal to x. 9-\frac{y}{2}=x Rearrange the equation to make y the subject. \begin{aligned} & 9-\frac{y}{2}=x \\\\ & 9-x=\frac{y}{2} \hspace{2cm} \text{Add } \frac{y}{2} \text{ to both sides and subtract } x \text{ from both sides. } \\\\ & 18-2x=y \hspace{1.8cm} \text{Multiply both sides by } 2. \end{aligned} Write your inverse function using the f^{-1} notation. Use g^{-1} because the original function was g. Example 3: Finding the inverse of a function involving a powerIf f(x)=2x^2-5 find f^{-1}(x) . Write out the expression for the original function using a y instead of the x . Set this expression equal to x. 2{{y}^{2}}-5=x Rearrange the equation to make y the subject. \begin{aligned} & 2{{y}^{2}}-5=x \\\\ & 2{{y}^{2}}=x+5 \hspace{3cm} \text{Add } 5 \text{ to both sides.. } \\\\ & {{y}^{2}}=\frac{x+5}{2} \hspace{3cm} \text{Divide both sides by } 2. \\\\ & y=\sqrt{\frac{x+5}{2}} \hspace{2.8cm} \text{Square root both sides.} \end{aligned}
Write your inverse function using the f^{-1} notation. {{f}^{-1}}\left( x \right)=\sqrt{\frac{x+5}{2}} Example 4: Finding the inverse of a function involving an algebraic fractionIf h(x)=\frac{x-3}{x+2} , find h^{-1}(x) . Write out the expression for the original function using a y instead of the x . Set this expression equal to x. \frac{y-3}{y+2}=x Rearrange the equation to make y the subject. \begin{aligned} & \frac{y-3}{y+2}=x \\\\ & y-3=x\left( y+2 \right) \hspace{2.7cm} \text{Multiply both sides by the denominator } y+2 \\\\ & y-3=xy+2x \hspace{2.8cm} \text{Expand the bracket} \\\\ & y-xy=2x+3 \hspace{2.8cm} \text{Isolate the } y \text{ by subtracting } xy \text{ from both sides}\\ & \hspace{5cm} \text{and adding } 3 \text{ to both sides} \\\\ & y\left( 1-x \right)=2x+3 \hspace{2.5cm} \text{Factor out the } y\\\\ & y=\frac{2x+3}{1-x} \hspace{3.4cm} \text{Divide both sides by } 1-x \end{aligned} Write your inverse function using the f^{-1} notation. {{h}^{-1}}\left( x \right)=\frac{2x+3}{1-x} Common misconceptions
A common error is to think the inverse function will be the reciprocal of the function because of the “power of -1 ” notation. E.g. Practice inverse function questionsRearranging the formula, first add 9, then divide by 4. Rearranging the formula, first subtract 7, then multiply by 3. Rearranging the formula, first add y to both sides, then subtract x from both sides. Rearranging the formula, first add 1, then cube root. Rearranging the formula, first subtract 6, then square and expand the brackets. See working and use example 4 to help. \begin{aligned} & \frac{2y}{y+3}=x \\\\ & 2y=x\left( y+3 \right) \\\\ & 2y=xy+3x \\\\ & 2y-xy=3x \\\\ & y\left( 2-x \right)=3x \\\\ & y=\frac{3x}{2-x} \\\\ & {{h}^{-1}}\left( x \right)=\frac{3x}{2-x} \end{aligned} Inverse functions GCSE questions1. (a) If f(x)=6x-5, find f^{-1}(x). (b) If g(x)=3+\sqrt{x}, find g^{-1}(x). (4 marks) Show answer (a) Attempt the rearrange formula 6y-5=x (1) \frac{x+5}{6} (1) (b) Attempt the rearrange formula 3+\sqrt{y}=x (1) (x-3)^2 or x^2-6x+9 (1) 2. If f(x)=3x-2 and g(x)=4x+7 (a) Find f^{-1}(4) (b) Find when f(x)=g^{-1}(x) (6 marks) Show answer (a) Set 3x-2=4 or find f^{-1}(x)=\frac{x+2}{3} (1) Answer of 2 (1) (b) g^{-1}(x)=\frac{x-7}{4} (1) Set equal 3x-2=\frac{x-7}{4} (1) Get x’s on one side of the equation 12x-x=-7+8 (1) Answer of x=\frac{1}{11} (1) 3. If f(x)=\frac{x+5}{x+3}, find f^{-1}(x). (4 marks) Show answer x=\frac{y+5}{y+3} x(y+3)=y+5 Multiplying by denominator (1) xy-y=5-3x Expand and isolate (1) y(x-1)=5-3x y factored out (1) f^{-1}(x)=\frac{5-3x}{x-1} (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 a oneDefinition: Inverse of a Function Defined by Ordered Pairs. If f(x) is a one-to-one function whose ordered pairs are of the form (x,y), then its inverse function f−1(x) is the set of ordered pairs (y,x).
What is an example of inverse of the oneLecture 1 : Inverse functions One-to-one Functions A function f is one-to-one if it never takes the same value twice or f(x1) = f(x2) whenever x1 = x2. Example The function f(x) = x is one to one, because if x1 = x2, then f(x1) = f(x2).
What is inverse function and worksheet?Inverse functions worksheets help students to understand the inverse function that undoes the effect of another function. For example, the inverse function of the formula that converts Celsius temperature to Fahrenheit temperature is the formula that converts Fahrenheit to Celsius.
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