method mark.","highlights":[{"text":"$$f(x - 2) = 0$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(x-2)^2-1=0"},{"latex":"x^2-4x+4-1=0"},{"latex":"x^2-4x+\\htmlData{anchor=constant_final}{3}=0"}],"highlights":[{"color":"sky","style":"box","anchor":"constant_final"}]},"order":1,"title":"Form a quadratic equation","description":"Expand the bracket and collect the constant terms. This gives the quadratic equation needed for the formula."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}"},{"latex":"x=\\frac{-(-4)\\pm\\sqrt{(-4)^2-4(1)(3)}}{2(1)}"},{"latex":"x=\\frac{4\\pm\\sqrt{4}}{2}"},{"latex":"x=\\frac{4+2}{2}=\\htmlData{anchor=plus_root_final}{3}"},{"latex":"x=\\frac{4-2}{2}=\\htmlData{anchor=minus_root_final}{1}"}],"highlights":[{"color":"green","style":"box","anchor":"plus_root_final"},{"color":"orange","style":"box","anchor":"minus_root_final"}]},"order":2,"title":"Apply the formula","description":"For this quadratic, a = 1, b = −4 and c = 3. Treat the plus and minus signs as two separate calculations."}]}],"generatedAt":"2026-08-16T10:12:15.098Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0fc3bce1-78bc-439a-ade4-ff39ca892037","revisionId":"5a1edef4-68f1-4aac-a476-da03aa6b71d1","questionNumber":"9","questionType":"LA","specification":"The function $f$ is defined such that $$f(x) = 4x - 1$$","difficulty":3,"difficultyBand":"hard","totalMarks":4,"sourceTitle":"Inverse and Composite Functions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"93174758-da3f-4605-af2d-1a819cef28bf","label":"a","content":"Find $f^{-1}(x)$","markscheme":"- $x = 4y - 1 \\Rightarrow x + 1 = 4y$ M1\n- $f^{-1}(x) = \\frac{x + 1}{4}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Attempts to rearrange $y = 4x - 1$.","code":"M"},{"id":"p2","marks":1,"description":"Correct expression $\\frac{x + 1}{4}$.","code":"A"}],"finalAnswers":["\\frac{x+1}{4}","0.25x+0.25"],"generatedAt":"2026-08-10T10:30:00.673Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"reverse-and-rearrange","label":"Reverse and rearrange","steps":[{"type":"text","order":0,"title":"Write the function as y","content":"Start by writing the output of the function as $y$:\n\n$$y=4x-1$$\n\nFor an inverse function, inputs and outputs exchange roles. So swap the letters $x$ and $y$ before rearranging.","highlights":[{"text":"Find $f^{-1}(x)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"x=4y-1"},{"latex":"x+1=4y"},{"latex":"y=\\dfrac{x+1}{\\htmlData{anchor=final_four}{4}}"}],"highlights":[{"color":"green","style":"box","anchor":"final_four"}]},"order":1,"title":"Swap and rearrange","description":"Swap the variables, then add 1 to both sides and divide both sides by 4 to make y the subject. This earns the method mark."},{"type":"text","order":2,"title":"State the inverse","content":"The rearranged $y$ gives the inverse function, so write it as:\n\n$$\\boxed{f^{-1}(x)=\\dfrac{x+1}{4}}$$\n\nThis earns the accuracy mark."},{"type":"text","order":3,"title":"State the inverse","content":"The rearranged $y$ gives the inverse function, so write it as:\n\n$$\\boxed{f^{-1}(x)=\\dfrac{x+1}{4}}$$\n\nThis earns the accuracy mark."}]}],"generatedAt":"2026-08-16T10:29:38.004Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]},{"id":"f997de42-8ca8-4649-a505-f71b32b4b3cc","label":"b","content":"The function $g$ is defined such that $g(x) = kx^2$ where $k$ is a constant. Given that $fg(2) = 12$, work out the value of $k$.","markscheme":"- $g(2) = 4k$ M1\n- $f(4k) = 4(4k) - 1 = 16k - 1$ M1\n- $16k - 1 = 12 \\Rightarrow 16k = 13 \\Rightarrow k = \\frac{13}{16}$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly finds $fg(2)$ in terms of $k$, e.g.

Inverse and Composite Functions

Question 26

Inverse and Composite Functions Questions

Inverse and Composite Functions

Question 26

Inverse and Composite Functions Questions