inverse of y=7-x^3

{ "query": { "display": "inverse $$y=7-x^{3}$$", "symbolab_question": "FUNCTION#inverse y=7-x^{3}" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "inverse", "default": "\\sqrt[3]{-x+7}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Inverse of $$7-x^{3}:{\\quad}\\sqrt[3]{-x+7}$$", "steps": [ { "type": "definition", "title": "Function Inverse definition", "text": "A function g is the inverse of function f if for $$y=f\\left(x\\right),\\:\\:x=g\\left(y\\right)\\:$$" }, { "type": "step", "result": "y=7-x^{3}" }, { "type": "interim", "title": "Replace $$x\\:$$with $$y$$", "input": "y=7-x^{3}", "result": "x=7-y^{3}", "steps": [ { "type": "step", "primary": "Replace $$x\\:$$with $$y$$", "secondary": [ "Replace $$y\\:$$with $$x$$" ], "result": "x=7-y^{3}" } ], "meta": { "interimType": "Interchange Variables 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vpqcDD/7yX5Tbd9fO9JFZcgW73PeTvX+ezS02rW5jrCODBqG0bRA4uqHPsV5PG7RRphnRUpbfJBjcr0sJwFNS0UqTd96MWTKI6Kr2Ib0iQCBEx4g4qN8zWEvXnzxyqLfLA8y6avxYBMhBIyqA2CEODiXOJb7o9HjRRcvm85H0Tc=" } }, { "type": "interim", "title": "Solve for $$y,\\:x=7-y^{3}$$", "input": "x=7-y^{3}", "steps": [ { "type": "step", "primary": "Switch sides", "result": "7-y^{3}=x" }, { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "7-y^{3}=x", "result": "-y^{3}=x-7", "steps": [ { "type": "step", "primary": "Subtract $$7$$ from both sides", "result": "7-y^{3}-7=x-7" }, { "type": "step", "primary": "Simplify", "result": "-y^{3}=x-7" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-y^{3}=x-7", "result": "y^{3}=-x+7", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-y^{3}}{-1}=\\frac{x}{-1}-\\frac{7}{-1}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-y^{3}}{-1}=\\frac{x}{-1}-\\frac{7}{-1}", "result": "y^{3}=-x+7", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-y^{3}}{-1}:{\\quad}y^{3}$$", "input": "\\frac{-y^{3}}{-1}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{y^{3}}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=y^{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WaGeGSM4dY2HAanlpXOx5zFSpzwRDPIFcbKdvhSPhgvMwViaLUXkeD+JukROhWdjNsF7bl9f0A7BbLLieJny3B429vuTSxWa7B/X3D1oP03AWQmX+FAZQ57eQ8HwbCJCzg2vc8v6O/P90WGKgenVtg==" } }, { "type": "interim", "title": "Simplify $$\\frac{x}{-1}-\\frac{7}{-1}:{\\quad}-x+7$$", "input": "\\frac{x}{-1}-\\frac{7}{-1}", "steps": [ { "type": "step", "primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{x-7}{-1}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{-b}=-\\frac{a}{b}$$", "result": "=-\\frac{x-7}{1}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{1}=a$$", "result": "=-\\left(x-7\\right)" }, { "type": "step", "primary": "Distribute parentheses", "result": "=-x-\\left(-7\\right)" }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a,\\:\\:\\:-\\left(a\\right)=-a$$" ], "result": "=-x+7" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Hv1l6HXHDerB0/+dW2WI3LuhEQ88o5MAwbnrtOQgUlDehkKrn0era9rz8TlL+x/vWro5nRDF75VO7IgYeP7K4qN6Hv6MoTMtvtU0IQwXdn9szOhN37mcRdV5CgGGkVwgd6QwF+awglKH7xM9d0X0hqbYXS+1BtlGfCt9IoPdTks=" } }, { "type": "step", "result": "y^{3}=-x+7" } ], "meta": { "interimType": "Generic Simplify 0Eq" } } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "step", "primary": "For $$x^n=f\\left(a\\right)$$, n is odd, the solution is $$x=\\sqrt[n]{f\\left(a\\right)}$$" }, { "type": "step", "result": "y=\\sqrt[3]{-x+7}" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve For Title 2Eq" } }, { "type": "step", "result": "\\sqrt[3]{-x+7}" } ], "meta": { "solvingClass": "Function Inverse" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "7-x^{3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }